:: Opposite Categories and Contravariant Functors
:: by Czes\l aw Byli\'nski
::
:: Received February 13, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users

:: Opposite Category
definition
let C be Category;
func C opp -> non empty non void strict CatStr equals :: OPPCAT_1:def 1
CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #);
coherence
CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #) is non empty non void strict CatStr
;
end;

:: deftheorem defines opp OPPCAT_1:def 1 :
for C being Category holds C opp = CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #);

definition
let C be Category;
let c be Object of C;
func c opp -> Object of (C opp) equals :: OPPCAT_1:def 2
c;
coherence
c is Object of (C opp)
;
end;

:: deftheorem defines opp OPPCAT_1:def 2 :
for C being Category
for c being Object of C holds c opp = c;

registration
let C be Category;
coherence
( C opp is Category-like & C opp is transitive & C opp is associative & C opp is reflexive & C opp is with_identities )
proof end;
end;

definition
let C be Category;
let c be Object of (C opp);
func opp c -> Object of C equals :: OPPCAT_1:def 3
c opp ;
coherence
c opp is Object of C
;
end;

:: deftheorem defines opp OPPCAT_1:def 3 :
for C being Category
for c being Object of (C opp) holds opp c = c opp ;

theorem :: OPPCAT_1:1
canceled;

::$CT theorem :: OPPCAT_1:2 for C being Category for c being Object of C holds (c opp) opp = c ; theorem :: OPPCAT_1:3 for C being Category for c being Object of C holds opp (c opp) = c ; theorem :: OPPCAT_1:4 for C being Category for c being Object of (C opp) holds (opp c) opp = c ; theorem Th4: :: OPPCAT_1:5 for C being Category for a, b being Object of C holds Hom (a,b) = Hom ((b opp),(a opp)) proof end; theorem Th5: :: OPPCAT_1:6 for C being Category for a, b being Object of (C opp) holds Hom (a,b) = Hom ((opp b),(opp a)) proof end; definition let C be Category; let f be Morphism of C; func f opp -> Morphism of (C opp) equals :: OPPCAT_1:def 4 f; coherence f is Morphism of (C opp) ; end; :: deftheorem defines opp OPPCAT_1:def 4 : for C being Category for f being Morphism of C holds f opp = f; definition let C be Category; let f be Morphism of (C opp); func opp f -> Morphism of C equals :: OPPCAT_1:def 5 f opp ; coherence f opp is Morphism of C ; end; :: deftheorem defines opp OPPCAT_1:def 5 : for C being Category for f being Morphism of (C opp) holds opp f = f opp ; definition let C be Category; let a, b be Object of C; assume A1: Hom (a,b) <> {} ; let f be Morphism of a,b; func f opp -> Morphism of b opp ,a opp equals :Def6: :: OPPCAT_1:def 6 f; coherence f is Morphism of b opp ,a opp proof end; end; :: deftheorem Def6 defines opp OPPCAT_1:def 6 : for C being Category for a, b being Object of C st Hom (a,b) <> {} holds for f being Morphism of a,b holds f opp = f; definition let C be Category; let a, b be Object of C; assume A1: Hom ((a opp),(b opp)) <> {} ; let f be Morphism of a opp ,b opp ; func opp f -> Morphism of b,a equals :Def7: :: OPPCAT_1:def 7 f; coherence f is Morphism of b,a proof end; end; :: deftheorem Def7 defines opp OPPCAT_1:def 7 : for C being Category for a, b being Object of C st Hom ((a opp),(b opp)) <> {} holds for f being Morphism of a opp ,b opp holds opp f = f; theorem :: OPPCAT_1:7 for C being Category for a, b being Object of C st Hom (a,b) <> {} holds for f being Morphism of a,b holds (f opp) opp = f proof end; theorem :: OPPCAT_1:8 for C being Category for a, b being Object of C st Hom (a,b) <> {} holds for f being Morphism of a,b holds opp (f opp) = f proof end; theorem :: OPPCAT_1:9 for C being Category for a, b being Object of (C opp) for f being Morphism of a,b holds (opp f) opp = f ; theorem Th9: :: OPPCAT_1:10 for C being Category for a, b being Object of C st Hom (a,b) <> {} holds for f being Morphism of a,b holds ( dom (f opp) = cod f & cod (f opp) = dom f ) proof end; theorem :: OPPCAT_1:11 for C being Category for a, b being Object of (C opp) for f being Morphism of a,b holds ( dom (opp f) = cod f & cod (opp f) = dom f ) ; theorem :: OPPCAT_1:12 for C being Category for a, b being Object of C st Hom (a,b) <> {} holds for f being Morphism of a,b holds ( (dom f) opp = cod (f opp) & (cod f) opp = dom (f opp) ) by Th9; theorem :: OPPCAT_1:13 for C being Category for a, b being Object of (C opp) st Hom (a,b) <> {} holds for f being Morphism of a,b holds ( opp (dom f) = cod (opp f) & opp (cod f) = dom (opp f) ) ; theorem :: OPPCAT_1:14 canceled; ::$CT
theorem Th13: :: OPPCAT_1:15
for C being Category
for a, b being Object of (C opp)
for f being Morphism of a,b st Hom (a,b) <> {} holds
opp f is Morphism of opp b, opp a
proof end;

theorem Th14: :: OPPCAT_1:16
for C being Category
for a, b, c being Object of C st Hom (a,b) <> {} & Hom (b,c) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds (g (*) f) opp = (f opp) (*) (g opp)
proof end;

theorem :: OPPCAT_1:17
for C being Category
for a, b, c being Object of C st Hom ((b opp),(a opp)) <> {} & Hom ((c opp),(b opp)) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds (g (*) f) opp = (f opp) (*) (g opp)
proof end;

theorem Th16: :: OPPCAT_1:18
for C being Category
for f, g being Morphism of (C opp) st dom g = cod f holds
opp (g (*) f) = (opp f) (*) (opp g)
proof end;

theorem :: OPPCAT_1:19
for C being Category
for a, b, c being Object of C
for f being Morphism of a,b
for g being Morphism of b,c st Hom (a,b) <> {} & Hom (b,c) <> {} holds
(g * f) opp = (f opp) (*) (g opp)
proof end;

Lm1: for C being Category
for a being Object of C
for b being Object of (C opp) holds
( ( Hom ((a opp),b) <> {} implies for f being Morphism of a opp ,b holds f (*) ((id a) opp) = f ) & ( Hom (b,(a opp)) <> {} implies for f being Morphism of b,a opp holds ((id a) opp) (*) f = f ) )

proof end;

theorem Th18: :: OPPCAT_1:20
for C being Category
for a being Object of C holds (id a) opp = id (a opp)
proof end;

Lm2: for C being Category
for a being Object of C holds id a = id (a opp)

proof end;

theorem Th19: :: OPPCAT_1:21
for C being Category
for a being Object of (C opp) holds opp (id a) = id (opp a)
proof end;

Lm3: for C being Category
for a, b, c being Object of C st Hom (a,b) <> {} & Hom (b,c) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds g * f = (f opp) * (g opp)

proof end;

theorem :: OPPCAT_1:22
for C being Category
for a, b being Object of C
for f being Morphism of a,b holds
( f opp is monic iff f is epi )
proof end;

theorem :: OPPCAT_1:23
for C being Category
for b, c being Object of C st Hom (b,c) <> {} holds
for f being Morphism of b,c holds
( f opp is epi iff f is monic )
proof end;

theorem :: OPPCAT_1:24
for C being Category
for a, b being Object of C
for f being Morphism of a,b holds
( f opp is invertible iff f is invertible )
proof end;

theorem :: OPPCAT_1:25
for C being Category
for c being Object of C holds
( c is initial iff c opp is terminal )
proof end;

theorem :: OPPCAT_1:26
for C being Category
for c being Object of C holds
( c opp is initial iff c is terminal )
proof end;

:: Contravariant Functors
definition
let C, B be Category;
let S be Function of the carrier' of (C opp), the carrier' of B;
func /* S -> Function of the carrier' of C, the carrier' of B means :Def8: :: OPPCAT_1:def 8
for f being Morphism of C holds it . f = S . (f opp);
existence
ex b1 being Function of the carrier' of C, the carrier' of B st
for f being Morphism of C holds b1 . f = S . (f opp)
proof end;
uniqueness
for b1, b2 being Function of the carrier' of C, the carrier' of B st ( for f being Morphism of C holds b1 . f = S . (f opp) ) & ( for f being Morphism of C holds b2 . f = S . (f opp) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def8 defines /* OPPCAT_1:def 8 :
for C, B being Category
for S being Function of the carrier' of (C opp), the carrier' of B
for b4 being Function of the carrier' of C, the carrier' of B holds
( b4 = /* S iff for f being Morphism of C holds b4 . f = S . (f opp) );

theorem :: OPPCAT_1:27
for B, C being Category
for S being Function of the carrier' of (C opp), the carrier' of B
for f being Morphism of (C opp) holds (/* S) . (opp f) = S . f
proof end;

Lm4: for B, C being Category
for S being Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj S) . (c opp))

proof end;

theorem Th26: :: OPPCAT_1:28
for B, C being Category
for S being Functor of C opp ,B
for c being Object of C holds (Obj (/* S)) . c = (Obj S) . (c opp)
proof end;

theorem :: OPPCAT_1:29
for B, C being Category
for S being Functor of C opp ,B
for c being Object of (C opp) holds (Obj (/* S)) . (opp c) = (Obj S) . c
proof end;

Lm5: for B, C being Category
for S being Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj (/* S)) . c)

proof end;

Lm6: now :: thesis: for C, B being Category
for S being Functor of C opp ,B
for c being Object of C ex d being Object of B st (/* S) . (id c) = id d
let C, B be Category; :: thesis: for S being Functor of C opp ,B
for c being Object of C ex d being Object of B st (/* S) . (id c) = id d

let S be Functor of C opp ,B; :: thesis: for c being Object of C ex d being Object of B st (/* S) . (id c) = id d
let c be Object of C; :: thesis: ex d being Object of B st (/* S) . (id c) = id d
(/* S) . (id c) = id ((Obj (/* S)) . c) by Lm5;
hence ex d being Object of B st (/* S) . (id c) = id d ; :: thesis: verum
end;

Lm7: for B, C being Category
for S being Functor of C opp ,B
for f being Morphism of C holds
( (Obj (/* S)) . (dom f) = cod ((/* S) . f) & (Obj (/* S)) . (cod f) = dom ((/* S) . f) )

proof end;

Lm8: now :: thesis: for C, B being Category
for S being Functor of C opp ,B
for f being Morphism of C holds
( (/* S) . (id (dom f)) = id (cod ((/* S) . f)) & (/* S) . (id (cod f)) = id (dom ((/* S) . f)) )
let C, B be Category; :: thesis: for S being Functor of C opp ,B
for f being Morphism of C holds
( (/* S) . (id (dom f)) = id (cod ((/* S) . f)) & (/* S) . (id (cod f)) = id (dom ((/* S) . f)) )

let S be Functor of C opp ,B; :: thesis: for f being Morphism of C holds
( (/* S) . (id (dom f)) = id (cod ((/* S) . f)) & (/* S) . (id (cod f)) = id (dom ((/* S) . f)) )

let f be Morphism of C; :: thesis: ( (/* S) . (id (dom f)) = id (cod ((/* S) . f)) & (/* S) . (id (cod f)) = id (dom ((/* S) . f)) )
thus (/* S) . (id (dom f)) = id ((Obj (/* S)) . (dom f)) by Lm5
.= id (cod ((/* S) . f)) by Lm7 ; :: thesis: (/* S) . (id (cod f)) = id (dom ((/* S) . f))
thus (/* S) . (id (cod f)) = id ((Obj (/* S)) . (cod f)) by Lm5
.= id (dom ((/* S) . f)) by Lm7 ; :: thesis: verum
end;

Lm9: for B, C being Category
for S being Functor of C opp ,B
for a, b, c being Object of C st Hom (a,b) <> {} & Hom (b,c) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds (/* S) . (g (*) f) = ((/* S) . f) (*) ((/* S) . g)

proof end;

definition
let C, D be Category;
mode Contravariant_Functor of C,D -> Function of the carrier' of C, the carrier' of D means :Def9: :: OPPCAT_1:def 9
( ( for c being Object of C ex d being Object of D st it . (id c) = id d ) & ( for f being Morphism of C holds
( it . (id (dom f)) = id (cod (it . f)) & it . (id (cod f)) = id (dom (it . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds
it . (g (*) f) = (it . f) (*) (it . g) ) );
existence
ex b1 being Function of the carrier' of C, the carrier' of D st
( ( for c being Object of C ex d being Object of D st b1 . (id c) = id d ) & ( for f being Morphism of C holds
( b1 . (id (dom f)) = id (cod (b1 . f)) & b1 . (id (cod f)) = id (dom (b1 . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds
b1 . (g (*) f) = (b1 . f) (*) (b1 . g) ) )
proof end;
end;

:: deftheorem Def9 defines Contravariant_Functor OPPCAT_1:def 9 :
for C, D being Category
for b3 being Function of the carrier' of C, the carrier' of D holds
( b3 is Contravariant_Functor of C,D iff ( ( for c being Object of C ex d being Object of D st b3 . (id c) = id d ) & ( for f being Morphism of C holds
( b3 . (id (dom f)) = id (cod (b3 . f)) & b3 . (id (cod f)) = id (dom (b3 . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds
b3 . (g (*) f) = (b3 . f) (*) (b3 . g) ) ) );

theorem Th28: :: OPPCAT_1:30
for C, D being Category
for S being Contravariant_Functor of C,D
for c being Object of C
for d being Object of D st S . (id c) = id d holds
(Obj S) . c = d
proof end;

theorem Th29: :: OPPCAT_1:31
for C, D being Category
for S being Contravariant_Functor of C,D
for c being Object of C holds S . (id c) = id ((Obj S) . c)
proof end;

theorem Th30: :: OPPCAT_1:32
for C, D being Category
for S being Contravariant_Functor of C,D
for f being Morphism of C holds
( (Obj S) . (dom f) = cod (S . f) & (Obj S) . (cod f) = dom (S . f) )
proof end;

theorem Th31: :: OPPCAT_1:33
for C, D being Category
for S being Contravariant_Functor of C,D
for f, g being Morphism of C st dom g = cod f holds
dom (S . f) = cod (S . g)
proof end;

theorem Th32: :: OPPCAT_1:34
for B, C being Category
for S being Functor of C opp ,B holds /* S is Contravariant_Functor of C,B
proof end;

theorem Th33: :: OPPCAT_1:35
for B, C, D being Category
for S1 being Contravariant_Functor of C,B
for S2 being Contravariant_Functor of B,D holds S2 * S1 is Functor of C,D
proof end;

Lm10: for B, C being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj S) . (c opp))

proof end;

theorem Th34: :: OPPCAT_1:36
for B, C being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of C holds (Obj (/* S)) . c = (Obj S) . (c opp)
proof end;

theorem :: OPPCAT_1:37
for B, C being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of (C opp) holds (Obj (/* S)) . (opp c) = (Obj S) . c
proof end;

Lm11: for B, C being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj (/* S)) . c)

proof end;

Lm12: for B, C being Category
for S being Contravariant_Functor of C opp ,B
for f being Morphism of C holds
( (Obj (/* S)) . (dom f) = dom ((/* S) . f) & (Obj (/* S)) . (cod f) = cod ((/* S) . f) )

proof end;

theorem :: OPPCAT_1:38
for B, C being Category
for S being Contravariant_Functor of C opp ,B holds /* S is Functor of C,B
proof end;

:: Dualization functors
definition
let C, B be Category;
let S be Function of the carrier' of C, the carrier' of B;
func *' S -> Function of the carrier' of (C opp), the carrier' of B means :Def10: :: OPPCAT_1:def 10
for f being Morphism of (C opp) holds it . f = S . (opp f);
existence
ex b1 being Function of the carrier' of (C opp), the carrier' of B st
for f being Morphism of (C opp) holds b1 . f = S . (opp f)
proof end;
uniqueness
for b1, b2 being Function of the carrier' of (C opp), the carrier' of B st ( for f being Morphism of (C opp) holds b1 . f = S . (opp f) ) & ( for f being Morphism of (C opp) holds b2 . f = S . (opp f) ) holds
b1 = b2
proof end;
func S *' -> Function of the carrier' of C, the carrier' of (B opp) means :Def11: :: OPPCAT_1:def 11
for f being Morphism of C holds it . f = (S . f) opp ;
existence
ex b1 being Function of the carrier' of C, the carrier' of (B opp) st
for f being Morphism of C holds b1 . f = (S . f) opp
proof end;
uniqueness
for b1, b2 being Function of the carrier' of C, the carrier' of (B opp) st ( for f being Morphism of C holds b1 . f = (S . f) opp ) & ( for f being Morphism of C holds b2 . f = (S . f) opp ) holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines *' OPPCAT_1:def 10 :
for C, B being Category
for S being Function of the carrier' of C, the carrier' of B
for b4 being Function of the carrier' of (C opp), the carrier' of B holds
( b4 = *' S iff for f being Morphism of (C opp) holds b4 . f = S . (opp f) );

:: deftheorem Def11 defines *' OPPCAT_1:def 11 :
for C, B being Category
for S being Function of the carrier' of C, the carrier' of B
for b4 being Function of the carrier' of C, the carrier' of (B opp) holds
( b4 = S *' iff for f being Morphism of C holds b4 . f = (S . f) opp );

theorem :: OPPCAT_1:39
for B, C being Category
for S being Function of the carrier' of C, the carrier' of B
for f being Morphism of C holds (*' S) . (f opp) = S . f
proof end;

Lm13: for B, C being Category
for S being Functor of C,B
for c being Object of (C opp) holds (*' S) . (id c) = id ((Obj S) . (opp c))

proof end;

theorem Th38: :: OPPCAT_1:40
for B, C being Category
for S being Functor of C,B
for c being Object of (C opp) holds (Obj (*' S)) . c = (Obj S) . (opp c)
proof end;

theorem :: OPPCAT_1:41
for B, C being Category
for S being Functor of C,B
for c being Object of C holds (Obj (*' S)) . (c opp) = (Obj S) . c
proof end;

Lm14: for B, C being Category
for S being Functor of C,B
for c being Object of C holds (S *') . (id c) = id (((Obj S) . c) opp)

proof end;

theorem Th40: :: OPPCAT_1:42
for B, C being Category
for S being Functor of C,B
for c being Object of C holds (Obj (S *')) . c = ((Obj S) . c) opp
proof end;

Lm15: for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of (C opp) holds (*' S) . (id c) = id ((Obj S) . (opp c))

proof end;

theorem Th41: :: OPPCAT_1:43
for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of (C opp) holds (Obj (*' S)) . c = (Obj S) . (opp c)
proof end;

theorem :: OPPCAT_1:44
for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (Obj (*' S)) . (c opp) = (Obj S) . c
proof end;

Lm16: for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (S *') . (id c) = id (((Obj S) . c) opp)

proof end;

theorem Th43: :: OPPCAT_1:45
for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (Obj (S *')) . c = ((Obj S) . c) opp
proof end;

Lm17: for C, D being Category
for F being Function of the carrier' of C, the carrier' of D
for f being Morphism of (C opp) holds ((*' F) *') . f = (F . (opp f)) opp

proof end;

theorem Th44: :: OPPCAT_1:46
for C, D being Category
for F being Function of the carrier' of C, the carrier' of D
for f being Morphism of C holds ((*' F) *') . (f opp) = (F . f) opp
proof end;

theorem Th45: :: OPPCAT_1:47
for C, D being Category
for S being Function of the carrier' of C, the carrier' of D holds /* (*' S) = S
proof end;

theorem :: OPPCAT_1:48
for C, D being Category
for S being Function of the carrier' of (C opp), the carrier' of D holds *' (/* S) = S
proof end;

theorem :: OPPCAT_1:49
for C, D being Category
for S being Function of the carrier' of C, the carrier' of D holds (*' S) *' = *' (S *')
proof end;

theorem :: OPPCAT_1:50
for C being Category
for D being strict Category
for S being Function of the carrier' of C, the carrier' of D holds (S *') *' = S
proof end;

theorem :: OPPCAT_1:51
for D being Category
for C being strict Category
for S being Function of the carrier' of C, the carrier' of D holds *' (*' S) = S
proof end;

Lm18: for B, C, D being Category
for S being Function of the carrier' of (C opp), the carrier' of B
for T being Function of the carrier' of B, the carrier' of D holds /* (T * S) = T * (/* S)

proof end;

theorem :: OPPCAT_1:52
for B, C, D being Category
for S being Function of the carrier' of C, the carrier' of B
for T being Function of the carrier' of B, the carrier' of D holds *' (T * S) = T * (*' S)
proof end;

theorem :: OPPCAT_1:53
for B, C, D being Category
for S being Function of the carrier' of C, the carrier' of B
for T being Function of the carrier' of B, the carrier' of D holds (T * S) *' = (T *') * S
proof end;

theorem :: OPPCAT_1:54
for B, C, D being Category
for F1 being Function of the carrier' of C, the carrier' of B
for F2 being Function of the carrier' of B, the carrier' of D holds (*' (F2 * F1)) *' = ((*' F2) *') * ((*' F1) *')
proof end;

Lm19: for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of (C opp) holds (*' S) . (id c) = id ((Obj (*' S)) . c)

proof end;

Lm20: for B, C being Category
for S being Contravariant_Functor of C,B
for f being Morphism of (C opp) holds
( (Obj (*' S)) . (dom f) = dom ((*' S) . f) & (Obj (*' S)) . (cod f) = cod ((*' S) . f) )

proof end;

theorem Th53: :: OPPCAT_1:55
for C, D being Category
for S being Contravariant_Functor of C,D holds *' S is Functor of C opp ,D
proof end;

Lm21: for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (S *') . (id c) = id ((Obj (S *')) . c)

proof end;

Lm22: for B, C being Category
for S being Contravariant_Functor of C,B
for f being Morphism of C holds
( (Obj (S *')) . (dom f) = dom ((S *') . f) & (Obj (S *')) . (cod f) = cod ((S *') . f) )

proof end;

theorem Th54: :: OPPCAT_1:56
for C, D being Category
for S being Contravariant_Functor of C,D holds S *' is Functor of C,D opp
proof end;

Lm23: for B, C being Category
for S being Functor of C,B
for c being Object of (C opp) holds (*' S) . (id c) = id ((Obj (*' S)) . c)

proof end;

Lm24: for B, C being Category
for S being Functor of C,B
for f being Morphism of (C opp) holds
( (Obj (*' S)) . (dom f) = cod ((*' S) . f) & (Obj (*' S)) . (cod f) = dom ((*' S) . f) )

proof end;

theorem Th55: :: OPPCAT_1:57
for C, D being Category
for S being Functor of C,D holds *' S is Contravariant_Functor of C opp ,D
proof end;

Lm25: for B, C being Category
for S being Functor of C,B
for c being Object of C holds (S *') . (id c) = id ((Obj (S *')) . c)

proof end;

Lm26: for B, C being Category
for S being Functor of C,B
for f being Morphism of C holds
( (Obj (S *')) . (dom f) = cod ((S *') . f) & (Obj (S *')) . (cod f) = dom ((S *') . f) )

proof end;

theorem Th56: :: OPPCAT_1:58
for C, D being Category
for S being Functor of C,D holds S *' is Contravariant_Functor of C,D opp
proof end;

theorem :: OPPCAT_1:59
for B, C, D being Category
for S1 being Contravariant_Functor of C,B
for S2 being Functor of B,D holds S2 * S1 is Contravariant_Functor of C,D
proof end;

theorem :: OPPCAT_1:60
for B, C, D being Category
for S1 being Functor of C,B
for S2 being Contravariant_Functor of B,D holds S2 * S1 is Contravariant_Functor of C,D
proof end;

theorem :: OPPCAT_1:61
for C, D being Category
for F being Functor of C,D
for c being Object of C holds (Obj ((*' F) *')) . (c opp) = ((Obj F) . c) opp
proof end;

theorem :: OPPCAT_1:62
for C, D being Category
for F being Contravariant_Functor of C,D
for c being Object of C holds (Obj ((*' F) *')) . (c opp) = ((Obj F) . c) opp
proof end;

theorem :: OPPCAT_1:63
for C, D being Category
for F being Functor of C,D holds (*' F) *' is Functor of C opp ,D opp
proof end;

theorem :: OPPCAT_1:64
for C, D being Category
for F being Contravariant_Functor of C,D holds (*' F) *' is Contravariant_Functor of C opp ,D opp
proof end;

:: Duality Functors
definition
let C be Category;
func id* C -> Contravariant_Functor of C,C opp equals :: OPPCAT_1:def 12
(id C) *' ;
coherence
(id C) *' is Contravariant_Functor of C,C opp
by Th56;
func *id C -> Contravariant_Functor of C opp ,C equals :: OPPCAT_1:def 13
*' (id C);
coherence
*' (id C) is Contravariant_Functor of C opp ,C
by Th55;
end;

:: deftheorem defines id* OPPCAT_1:def 12 :
for C being Category holds id* C = (id C) *' ;

:: deftheorem defines *id OPPCAT_1:def 13 :
for C being Category holds *id C = *' (id C);

theorem Th63: :: OPPCAT_1:65
for C being Category
for f being Morphism of C holds (id* C) . f = f opp
proof end;

theorem :: OPPCAT_1:66
for C being Category
for c being Object of C holds (Obj (id* C)) . c = c opp
proof end;

theorem Th65: :: OPPCAT_1:67
for C being Category
for f being Morphism of (C opp) holds (*id C) . f = opp f
proof end;

theorem :: OPPCAT_1:68
for C being Category
for c being Object of (C opp) holds (Obj (*id C)) . c = opp c
proof end;

theorem :: OPPCAT_1:69
for C, D being Category
for S being Function of the carrier' of C, the carrier' of D holds
( *' S = S * (*id C) & S *' = (id* D) * S )
proof end;

theorem :: OPPCAT_1:70
for C being Category
for a, b, c being Object of C st Hom (a,b) <> {} & Hom (b,c) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds g * f = (f opp) * (g opp) by Lm3;

theorem Th69: :: OPPCAT_1:71
for C being Category
for a being Object of C holds id a = id (a opp) by Lm2;

theorem :: OPPCAT_1:72
for C being Category
for a being Object of (C opp) holds id a = id (opp a)
proof end;