let A, B be non empty transitive with_units AltCatStr ; :: thesis: for F being contravariant 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 coretraction
let F be contravariant 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 coretraction
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 coretraction
let a be Morphism of o1,o2; :: thesis: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction implies F . a is coretraction )
assume A1: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} ) ; :: thesis: ( not a is retraction or F . a is coretraction )
assume a is retraction ; :: thesis: F . a is coretraction
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 3 :: thesis: F . b is_left_inverse_of F . a
a * b = idm o2 by A2;
hence (F . b) * (F . a) = F . (idm o2) by A1, FUNCTOR0:def 24
.= idm (F . o2) by Th13 ;
:: according to ALTCAT_3:def 1 :: thesis: verum