Assume I have $f : A \to B$, $g_a : A \to A$, $g_b : B \to B$ and $R$ an equivalence relation on $B$. If I know $f(a)$ is $R$-related to $g_b (f (g_a (a)))$ for all $a\in A$, is it the case that there must exist an $h$ where $h(a) = g_b(h(g_a(a)))$? (That is, I have a recurrence for $f$ in terms of the equivalence. Is there necessarily then a recurrence with respect to equality that uses the same equational form?)
I appreciate that I could quotient $B$ by $R$ and define an $h'$ that had the “right” equation over the quotiented $B$ (perhaps this relies on $g_b$ being well-behaved), but I want an equation on the original underlying set $B$.
If it’s not generally true, under what conditions is it true?