let A be category; :: thesis: for B being non empty subcategory of A holds B,B opp are_anti-isomorphic_under dualizing-func (A,(A opp))

let B be non empty subcategory of A; :: thesis: B,B opp are_anti-isomorphic_under dualizing-func (A,(A opp))

set F = dualizing-func (A,(A opp));

A1: B,B opp are_opposite by YELLOW18:def 4;

thus ( B is subcategory of A & B opp is subcategory of A opp ) by Th48; :: according to YELLOW20:def 5 :: thesis: ex G being contravariant Functor of B,B opp st

( G is bijective & ( for a9 being Object of B

for a being Object of A st a9 = a holds

G . a9 = (dualizing-func (A,(A opp))) . a ) & ( for b9, c9 being Object of B

for b, c being Object of A st <^b9,c9^> <> {} & b9 = b & c9 = c holds

for f9 being Morphism of b9,c9

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f ) )

take G = dualizing-func (B,(B opp)); :: thesis: ( G is bijective & ( for a9 being Object of B

for a being Object of A st a9 = a holds

G . a9 = (dualizing-func (A,(A opp))) . a ) & ( for b9, c9 being Object of B

for b, c being Object of A st <^b9,c9^> <> {} & b9 = b & c9 = c holds

for f9 being Morphism of b9,c9

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f ) )

thus G is bijective ; :: thesis: ( ( for a9 being Object of B

for a being Object of A st a9 = a holds

G . a9 = (dualizing-func (A,(A opp))) . a ) & ( for b9, c9 being Object of B

for b, c being Object of A st <^b9,c9^> <> {} & b9 = b & c9 = c holds

for f9 being Morphism of b9,c9

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f ) )

A2: A,A opp are_opposite by YELLOW18:def 4;

for f9 being Morphism of b,c

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f

let b1, c1 be Object of A; :: thesis: ( <^b,c^> <> {} & b = b1 & c = c1 implies for f9 being Morphism of b,c

for f being Morphism of b1,c1 st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f )

assume that

A3: <^b,c^> <> {} and

A4: ( b = b1 & c = c1 ) ; :: thesis: for f9 being Morphism of b,c

for f being Morphism of b1,c1 st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f

let f be Morphism of b,c; :: thesis: for f being Morphism of b1,c1 st f = f holds

G . f = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f

let f1 be Morphism of b1,c1; :: thesis: ( f = f1 implies G . f = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f1 )

assume A5: f = f1 ; :: thesis: G . f = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f1

A6: ( <^b,c^> c= <^b1,c1^> & f in <^b,c^> ) by A3, A4, ALTCAT_2:31;

then A7: <^((dualizing-func (A,(A opp))) . c1),((dualizing-func (A,(A opp))) . b1)^> <> {} by FUNCTOR0:def 19;

thus G . f = f by A1, A3, YELLOW18:def 5

.= (dualizing-func (A,(A opp))) . f1 by A2, A5, A6, YELLOW18:def 5

.= (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f1 by A6, A7, FUNCTOR0:def 16 ; :: thesis: verum

let B be non empty subcategory of A; :: thesis: B,B opp are_anti-isomorphic_under dualizing-func (A,(A opp))

set F = dualizing-func (A,(A opp));

A1: B,B opp are_opposite by YELLOW18:def 4;

thus ( B is subcategory of A & B opp is subcategory of A opp ) by Th48; :: according to YELLOW20:def 5 :: thesis: ex G being contravariant Functor of B,B opp st

( G is bijective & ( for a9 being Object of B

for a being Object of A st a9 = a holds

G . a9 = (dualizing-func (A,(A opp))) . a ) & ( for b9, c9 being Object of B

for b, c being Object of A st <^b9,c9^> <> {} & b9 = b & c9 = c holds

for f9 being Morphism of b9,c9

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f ) )

take G = dualizing-func (B,(B opp)); :: thesis: ( G is bijective & ( for a9 being Object of B

for a being Object of A st a9 = a holds

G . a9 = (dualizing-func (A,(A opp))) . a ) & ( for b9, c9 being Object of B

for b, c being Object of A st <^b9,c9^> <> {} & b9 = b & c9 = c holds

for f9 being Morphism of b9,c9

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f ) )

thus G is bijective ; :: thesis: ( ( for a9 being Object of B

for a being Object of A st a9 = a holds

G . a9 = (dualizing-func (A,(A opp))) . a ) & ( for b9, c9 being Object of B

for b, c being Object of A st <^b9,c9^> <> {} & b9 = b & c9 = c holds

for f9 being Morphism of b9,c9

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f ) )

A2: A,A opp are_opposite by YELLOW18:def 4;

hereby :: thesis: for b9, c9 being Object of B

for b, c being Object of A st <^b9,c9^> <> {} & b9 = b & c9 = c holds

for f9 being Morphism of b9,c9

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f

let b, c be Object of B; :: thesis: for b, c being Object of A st <^b,c^> <> {} & b = b & c = c holds for b, c being Object of A st <^b9,c9^> <> {} & b9 = b & c9 = c holds

for f9 being Morphism of b9,c9

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f

let a be Object of B; :: thesis: for a1 being Object of A st a = a1 holds

G . a = (dualizing-func (A,(A opp))) . a1

let a1 be Object of A; :: thesis: ( a = a1 implies G . a = (dualizing-func (A,(A opp))) . a1 )

assume a = a1 ; :: thesis: G . a = (dualizing-func (A,(A opp))) . a1

hence G . a = a1 by A1, YELLOW18:def 5

.= (dualizing-func (A,(A opp))) . a1 by A2, YELLOW18:def 5 ;

:: thesis: verum

end;G . a = (dualizing-func (A,(A opp))) . a1

let a1 be Object of A; :: thesis: ( a = a1 implies G . a = (dualizing-func (A,(A opp))) . a1 )

assume a = a1 ; :: thesis: G . a = (dualizing-func (A,(A opp))) . a1

hence G . a = a1 by A1, YELLOW18:def 5

.= (dualizing-func (A,(A opp))) . a1 by A2, YELLOW18:def 5 ;

:: thesis: verum

for f9 being Morphism of b,c

for f being Morphism of b,c st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b,c)) . f

let b1, c1 be Object of A; :: thesis: ( <^b,c^> <> {} & b = b1 & c = c1 implies for f9 being Morphism of b,c

for f being Morphism of b1,c1 st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f )

assume that

A3: <^b,c^> <> {} and

A4: ( b = b1 & c = c1 ) ; :: thesis: for f9 being Morphism of b,c

for f being Morphism of b1,c1 st f9 = f holds

G . f9 = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f

let f be Morphism of b,c; :: thesis: for f being Morphism of b1,c1 st f = f holds

G . f = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f

let f1 be Morphism of b1,c1; :: thesis: ( f = f1 implies G . f = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f1 )

assume A5: f = f1 ; :: thesis: G . f = (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f1

A6: ( <^b,c^> c= <^b1,c1^> & f in <^b,c^> ) by A3, A4, ALTCAT_2:31;

then A7: <^((dualizing-func (A,(A opp))) . c1),((dualizing-func (A,(A opp))) . b1)^> <> {} by FUNCTOR0:def 19;

thus G . f = f by A1, A3, YELLOW18:def 5

.= (dualizing-func (A,(A opp))) . f1 by A2, A5, A6, YELLOW18:def 5

.= (Morph-Map ((dualizing-func (A,(A opp))),b1,c1)) . f1 by A6, A7, FUNCTOR0:def 16 ; :: thesis: verum