I'm searching for the name of a category of similarity between functions; which takes the following form:
Suppose for some pair of surjective functions $f: X\rightarrow Y \subseteq X$, $g: A \rightarrow B \subseteq A$.
Suppose also that $h: X\rightarrow A$ and $i: Y\rightarrow B$ are bijective and total.
The following must hold true:
$f(x)=y \iff g(h(x)) = i(y)$ and conversely $g(a)=b \iff f(h^{-1}(a)) = i^{-1}(b)$
Is there a name for a more arbitrary type of equivalence which holds over functions with arbitrary sets as their domain?