let A, B be category; :: 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 iso holds

F . a is iso

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 iso holds

F . a is iso

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

F . a is iso

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

assume that

A1: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} ) and

A2: a is iso ; :: thesis: F . a is iso

( a is retraction & a is coretraction ) by A1, A2, ALTCAT_3:6;

then A3: ( F . a is retraction & F . a is coretraction ) by A1, Th18, Th19;

( <^(F . o1),(F . o2)^> <> {} & <^(F . o2),(F . o1)^> <> {} ) by A1, FUNCTOR0:def 18;

hence F . a is iso by A3, ALTCAT_3:6; :: thesis: verum

for o1, o2 being Object of A

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

F . a is iso

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 iso holds

F . a is iso

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

F . a is iso

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

assume that

A1: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} ) and

A2: a is iso ; :: thesis: F . a is iso

( a is retraction & a is coretraction ) by A1, A2, ALTCAT_3:6;

then A3: ( F . a is retraction & F . a is coretraction ) by A1, Th18, Th19;

( <^(F . o1),(F . o2)^> <> {} & <^(F . o2),(F . o1)^> <> {} ) by A1, FUNCTOR0:def 18;

hence F . a is iso by A3, ALTCAT_3:6; :: thesis: verum