A3:
F is_naturally_transformable_to F2
by A1, A2, Th19;
A4:
F1 is_transformable_to F2
by A2;
A5:
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (t2 . b) * (F1 /. f) = (F2 /. f) * (t2 . a)
by A2, Def7;
A6:
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (t1 . b) * (F /. f) = (F1 /. f) * (t1 . a)
by A1, Def7;
F is_transformable_to F1
by A1;
then
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds ((t2 `*` t1) . b) * (F /. f) = (F2 /. f) * ((t2 `*` t1) . a)
by A4, A6, A5, Lm2;
hence
t2 `*` t1 is natural_transformation of F,F2
by A3, Def7; verum