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 F1
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 F1
let e be natural_equivalence of F1,F2; :: thesis: ( F1,F2 are_naturally_equivalent implies (e ") `*` e = idt F1 )
assume A1: F1,F2 are_naturally_equivalent ; :: thesis: (e ") `*` e = idt F1
then A2: ( F1 is_transformable_to F2 & F2 is_transformable_to F1 ) by Def4;
A3: ( F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to F1 ) by A1, Def4;
now :: thesis: for a being Object of A holds ((e ") `*` e) ! a = (idt F1) ! a
let a be Object of A; :: thesis: ((e ") `*` e) ! a = (idt F1) ! a
A4: e ! a is iso by A1, Def5;
thus ((e ") `*` e) ! a = ((e ") `*` e) ! a by A3, FUNCTOR2:def 8
.= ((e ") ! a) * (e ! a) by A2, FUNCTOR2:def 5
.= ((e ! a) ") * (e ! a) by A1, Th38
.= idm (F1 . a) by A4, ALTCAT_3:def 5
.= (idt F1) ! a by FUNCTOR2:4 ; :: thesis: verum
end;
hence (e ") `*` e = idt F1 by FUNCTOR2:3; :: thesis: verum