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