What's the correct term for it?

[zef]

How do we call such relation between mappings f:A->A and g:B->B when there exists one-to-one mapping h:A->B such that for any a from A f(a)=g(h(a))?

Example.

f: {T, F) -> {T, F}
where f(x) = not x

g : {0, 1} -> {0, 1}
where f(x)=1-x

h: {T, F} -> {0, 1}
where h(T)=1, h(F)=0

 

[TenaliRaman]

Hmm, technically f(a) cannot be equal to g(h(a)) unless codomain(f) = codomain(g).

 

[HallsofIvy]

You just said f:A->A and g:B->B. For any function h:A->B, g(h(x)) is in B, not A.

 

[zef]

Sorry, it should be h(f(a))=g(h(a))

 

[HallsofIvy]

I don't know any specific name for the function h, but what you have is a "commutative diagram" used in Algebraic Topology and Category Theory.