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 " ) = idt F2
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 " ) = idt F2
let e be natural_equivalence of F1,F2; :: thesis: ( F1,F2 are_naturally_equivalent implies e `*` (e " ) = idt F2 )
assume A1: F1,F2 are_naturally_equivalent ; :: thesis: e `*` (e " ) = idt F2
then A2: F1 is_naturally_transformable_to F2 by Def4;
A3: F1 is_transformable_to F2 by A1, Def4;
A4: F2 is_naturally_transformable_to F1 by A1, Def4;
A5: F2 is_transformable_to F1 by A1, Def4;

now

let a be object of A; :: thesis: (e `*` (e " )) ! a = (idt F2) ! a
A6: e ! a is iso by A1, Def5;
thus (e `*` (e " )) ! a = (e `*` (e " )) ! a by A2, A4, FUNCTOR2:def 8
.= (e ! a) * ((e " ) ! a) by A3, A5, FUNCTOR2:def 5
.= (e ! a) * ((e ! a) " ) by A1, Th38
.= idm (F2 . a) by A6, ALTCAT_3:def 5
.= (idt F2) ! a by FUNCTOR2:6 ; :: thesis: verum

end;

hence e `*` (e " ) = idt F2 by FUNCTOR2:5; :: thesis: verum