let A, B be non empty transitive with_units AltCatStr ; :: thesis: for F1, F2 being Functor of A,B st F1 is_transformable_to F2 holds
for t being transformation of F1,F2 holds
( (idt F2) `*` t = t & t `*` (idt F1) = t )
let F1, F2 be Functor of A,B; :: thesis: ( F1 is_transformable_to F2 implies for t being transformation of F1,F2 holds
( (idt F2) `*` t = t & t `*` (idt F1) = t ) )
assume A1: F1 is_transformable_to F2 ; :: thesis: for t being transformation of F1,F2 holds
( (idt F2) `*` t = t & t `*` (idt F1) = t )
let t be transformation of F1,F2; :: thesis: ( (idt F2) `*` t = t & t `*` (idt F1) = t )
now :: thesis: for a being Object of A holds ((idt F2) `*` t) ! a = t ! a
let a be Object of A; :: thesis: ((idt F2) `*` t) ! a = t ! a
A2: <^(F1 . a),(F2 . a)^> <> {} by A1;
thus ((idt F2) `*` t) ! a = ((idt F2) ! a) * (t ! a) by A1, Def5
.= (idm (F2 . a)) * (t ! a) by Th4
.= t ! a by A2, ALTCAT_1:20 ; :: thesis: verum
end;
hence (idt F2) `*` t = t by A1, Th3; :: thesis: t `*` (idt F1) = t
now :: thesis: for a being Object of A holds (t `*` (idt F1)) ! a = t ! a
let a be Object of A; :: thesis: (t `*` (idt F1)) ! a = t ! a
A3: <^(F1 . a),(F2 . a)^> <> {} by A1;
thus (t `*` (idt F1)) ! a = (t ! a) * ((idt F1) ! a) by A1, Def5
.= (t ! a) * (idm (F1 . a)) by Th4
.= t ! a by A3, ALTCAT_1:def 17 ; :: thesis: verum
end;
hence t `*` (idt F1) = t by A1, Th3; :: thesis: verum