let A, B be non empty transitive with_units AltCatStr ; :: thesis: for F being covariant Functor of A,B

for o1, o2 being Object of A

for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction holds

F . a is retraction

let F be covariant Functor of A,B; :: thesis: for o1, o2 being Object of A

for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction holds

F . a is retraction

let o1, o2 be Object of A; :: thesis: for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction holds

F . a is retraction

let a be Morphism of o1,o2; :: thesis: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction implies F . a is retraction )

assume A1: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} ) ; :: thesis: ( not a is retraction or F . a is retraction )

assume a is retraction ; :: thesis: F . a is retraction

then consider b being Morphism of o2,o1 such that

A2: b is_right_inverse_of a ;

take F . b ; :: according to ALTCAT_3:def 2 :: thesis: F . a is_left_inverse_of F . b

a * b = idm o2 by A2;

hence (F . a) * (F . b) = F . (idm o2) by A1, FUNCTOR0:def 23

.= idm (F . o2) by FUNCTOR2:1 ;

:: according to ALTCAT_3:def 1 :: thesis: verum

for o1, o2 being Object of A

for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction holds

F . a is retraction

let F be covariant Functor of A,B; :: thesis: for o1, o2 being Object of A

for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction holds

F . a is retraction

let o1, o2 be Object of A; :: thesis: for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction holds

F . a is retraction

let a be Morphism of o1,o2; :: thesis: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction implies F . a is retraction )

assume A1: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} ) ; :: thesis: ( not a is retraction or F . a is retraction )

assume a is retraction ; :: thesis: F . a is retraction

then consider b being Morphism of o2,o1 such that

A2: b is_right_inverse_of a ;

take F . b ; :: according to ALTCAT_3:def 2 :: thesis: F . a is_left_inverse_of F . b

a * b = idm o2 by A2;

hence (F . a) * (F . b) = F . (idm o2) by A1, FUNCTOR0:def 23

.= idm (F . o2) by FUNCTOR2:1 ;

:: according to ALTCAT_3:def 1 :: thesis: verum