let F, G be strict covariant Functor of A, Concretized A; :: thesis: ( ( for a being object of A holds F . a = a ) & ( for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds (F . f) . [(idm a),[a,a]] = [f,[a,b]] ) & ( for a being object of A holds G . a = a ) & ( for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds (G . f) . [(idm a),[a,a]] = [f,[a,b]] ) implies F = G )
assume that
A1: for a being object of A holds F . a = a and
A2: for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds (F . f) . [(idm a),[a,a]] = [f,[a,b]] and
A3: for a being object of A holds G . a = a and
A4: for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds (G . f) . [(idm a),[a,a]] = [f,[a,b]] ; :: thesis: F = G

A5: now

let a be object of A; :: thesis: F . a = G . a
thus F . a = a by A1
.= G . a by A3 ; :: thesis: verum

end;

now

let a, b be object of A; :: thesis: ( <^a,b^> <> {} implies for f being Morphism of a,b holds F . f = G . f )
assume A6: <^a,b^> <> {} ; :: thesis: for f being Morphism of a,b holds F . f = G . f
let f be Morphism of a,b; :: thesis: F . f = G . f
A7: (F . f) . [(idm a),[a,a]] = [f,[a,b]] by A2, A6;
A8: (G . f) . [(idm a),[a,a]] = [f,[a,b]] by A4, A6;
( <^(F . a),(F . b)^> <> {} & F . a = a & F . b = b & G . a = a & G . b = b ) by A1, A3, A6, FUNCTOR0:def 19;
hence F . f = G . f by A6, A7, A8, Th47; :: thesis: verum

end;

hence F = G by A5, Th1; :: thesis: verum