let A, B be category; :: thesis: for F1, F2 being covariant Functor of A,B
for e being natural_equivalence of F1,F2 st F1,F2 are_naturally_equivalent holds
(e ") " = e
let F1, F2 be covariant Functor of A,B; :: thesis: for e being natural_equivalence of F1,F2 st F1,F2 are_naturally_equivalent holds
(e ") " = e
let e be natural_equivalence of F1,F2; :: thesis: ( F1,F2 are_naturally_equivalent implies (e ") " = e )
assume A1: F1,F2 are_naturally_equivalent ; :: thesis: (e ") " = e
then A2: F1 is_transformable_to F2 by Def4;

now :: thesis: for a being Object of A holds ((e ") ") ! a = e ! a

let a be Object of A; :: thesis: ((e ") ") ! a = e ! a
A3: <^(F1 . a),(F2 . a)^> <> {} by A2;
F2 is_transformable_to F1 by A1;
then A4: <^(F2 . a),(F1 . a)^> <> {} ;
e ! a is iso by A1, Def5;
then A5: ( e ! a is retraction & e ! a is coretraction ) by ALTCAT_3:5;
thus ((e ") ") ! a = ((e ") ! a) " by A1, Th38
.= ((e ! a) ") " by A1, Th38
.= e ! a by A3, A4, A5, ALTCAT_3:3 ; :: thesis: verum

end;

hence (e ") " = e by A2, FUNCTOR2:3; :: thesis: verum