let A, B be non empty AltGraph ; :: thesis: for F being Covariant FunctorStr over A,B st F is surjective holds

for a, b being Object of B st <^a,b^> <> {} holds

for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

let F be Covariant FunctorStr over A,B; :: thesis: ( F is surjective implies for a, b being Object of B st <^a,b^> <> {} holds

for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g ) )

given f being ManySortedFunction of the Arrows of A, the Arrows of B * the ObjectMap of F such that A1: ( f = the MorphMap of F & f is "onto" ) ; :: according to FUNCTOR0:def 32,FUNCTOR0:def 34 :: thesis: ( not F is onto or for a, b being Object of B st <^a,b^> <> {} holds

for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g ) )

assume A2: rng the ObjectMap of F = [: the carrier of B, the carrier of B:] ; :: according to FUNCTOR0:def 7,FUNCT_2:def 3 :: thesis: for a, b being Object of B st <^a,b^> <> {} holds

for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

let a, b be Object of B; :: thesis: ( <^a,b^> <> {} implies for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g ) )

assume A3: <^a,b^> <> {} ; :: thesis: for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

the ObjectMap of F is Covariant by FUNCTOR0:def 12;

then consider g being Function of the carrier of A, the carrier of B such that

A4: the ObjectMap of F = [:g,g:] ;

let f be Morphism of a,b; :: thesis: ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

( dom the ObjectMap of F = [: the carrier of A, the carrier of A:] & [a,b] in rng the ObjectMap of F ) by A2, FUNCT_2:def 1, ZFMISC_1:def 2;

then consider x being object such that

A5: x in [: the carrier of A, the carrier of A:] and

A6: [a,b] = the ObjectMap of F . x by FUNCT_1:def 3;

consider c, d being object such that

A7: ( c in the carrier of A & d in the carrier of A ) and

A8: [c,d] = x by A5, ZFMISC_1:def 2;

reconsider c = c, d = d as Object of A by A7;

the ObjectMap of F . (d,d) = [(g . d),(g . d)] by A4, FUNCT_3:75;

then A9: F . d = g . d ;

the ObjectMap of F . (c,c) = [(g . c),(g . c)] by A4, FUNCT_3:75;

then F . c = g . c ;

then A10: the ObjectMap of F . (c,d) = [(F . c),(F . d)] by A4, A9, FUNCT_3:75;

rng (Morph-Map (F,c,d)) = ( the Arrows of B * the ObjectMap of F) . [c,d] by A1, A5, A8

.= <^a,b^> by A5, A6, A8, FUNCT_2:15 ;

then consider g being object such that

A11: g in dom (Morph-Map (F,c,d)) and

A12: f = (Morph-Map (F,c,d)) . g by A3, FUNCT_1:def 3;

take c ; :: thesis: ex d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

take d ; :: thesis: ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

reconsider g = g as Morphism of c,d by A11;

take g ; :: thesis: ( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

thus ( a = F . c & b = F . d & <^c,d^> <> {} ) by A6, A8, A10, A11, XTUPLE_0:1; :: thesis: f = F . g

thus f = F . g by A3, A6, A8, A10, A11, A12, FUNCTOR0:def 15; :: thesis: verum

for a, b being Object of B st <^a,b^> <> {} holds

for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

let F be Covariant FunctorStr over A,B; :: thesis: ( F is surjective implies for a, b being Object of B st <^a,b^> <> {} holds

for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g ) )

given f being ManySortedFunction of the Arrows of A, the Arrows of B * the ObjectMap of F such that A1: ( f = the MorphMap of F & f is "onto" ) ; :: according to FUNCTOR0:def 32,FUNCTOR0:def 34 :: thesis: ( not F is onto or for a, b being Object of B st <^a,b^> <> {} holds

for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g ) )

assume A2: rng the ObjectMap of F = [: the carrier of B, the carrier of B:] ; :: according to FUNCTOR0:def 7,FUNCT_2:def 3 :: thesis: for a, b being Object of B st <^a,b^> <> {} holds

for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

let a, b be Object of B; :: thesis: ( <^a,b^> <> {} implies for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g ) )

assume A3: <^a,b^> <> {} ; :: thesis: for f being Morphism of a,b ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

the ObjectMap of F is Covariant by FUNCTOR0:def 12;

then consider g being Function of the carrier of A, the carrier of B such that

A4: the ObjectMap of F = [:g,g:] ;

let f be Morphism of a,b; :: thesis: ex c, d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

( dom the ObjectMap of F = [: the carrier of A, the carrier of A:] & [a,b] in rng the ObjectMap of F ) by A2, FUNCT_2:def 1, ZFMISC_1:def 2;

then consider x being object such that

A5: x in [: the carrier of A, the carrier of A:] and

A6: [a,b] = the ObjectMap of F . x by FUNCT_1:def 3;

consider c, d being object such that

A7: ( c in the carrier of A & d in the carrier of A ) and

A8: [c,d] = x by A5, ZFMISC_1:def 2;

reconsider c = c, d = d as Object of A by A7;

the ObjectMap of F . (d,d) = [(g . d),(g . d)] by A4, FUNCT_3:75;

then A9: F . d = g . d ;

the ObjectMap of F . (c,c) = [(g . c),(g . c)] by A4, FUNCT_3:75;

then F . c = g . c ;

then A10: the ObjectMap of F . (c,d) = [(F . c),(F . d)] by A4, A9, FUNCT_3:75;

rng (Morph-Map (F,c,d)) = ( the Arrows of B * the ObjectMap of F) . [c,d] by A1, A5, A8

.= <^a,b^> by A5, A6, A8, FUNCT_2:15 ;

then consider g being object such that

A11: g in dom (Morph-Map (F,c,d)) and

A12: f = (Morph-Map (F,c,d)) . g by A3, FUNCT_1:def 3;

take c ; :: thesis: ex d being Object of A ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

take d ; :: thesis: ex g being Morphism of c,d st

( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

reconsider g = g as Morphism of c,d by A11;

take g ; :: thesis: ( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )

thus ( a = F . c & b = F . d & <^c,d^> <> {} ) by A6, A8, A10, A11, XTUPLE_0:1; :: thesis: f = F . g

thus f = F . g by A3, A6, A8, A10, A11, A12, FUNCTOR0:def 15; :: thesis: verum