let A, B be Category; :: thesis: for F1, F2 being Functor of A,B st F1 is_transformable_to F2 holds
for t being transformation of F1,F2 st t is invertible holds
for a being Object of A holds F1 . a,F2 . a are_isomorphic
let F1, F2 be Functor of A,B; :: thesis: ( F1 is_transformable_to F2 implies for t being transformation of F1,F2 st t is invertible holds
for a being Object of A holds F1 . a,F2 . a are_isomorphic )
assume A1: F1 is_transformable_to F2 ; :: thesis: for t being transformation of F1,F2 st t is invertible holds
for a being Object of A holds F1 . a,F2 . a are_isomorphic
let t be transformation of F1,F2; :: thesis: ( t is invertible implies for a being Object of A holds F1 . a,F2 . a are_isomorphic )
assume A2: t is invertible ; :: thesis: for a being Object of A holds F1 . a,F2 . a are_isomorphic
let a be Object of A; :: thesis: F1 . a,F2 . a are_isomorphic
thus Hom (F1 . a),(F2 . a) <> {} by A1, NATTRA_1:def 2; :: according to CAT_1:def 17 :: thesis: ex b₁ being Morphism of F1 . a,F2 . a st b₁ is invertible
t . a is invertible by A2, NATTRA_1:def 10;
hence ex b₁ being Morphism of F1 . a,F2 . a st b₁ is invertible ; :: thesis: verum