I have the function $f:A\times B \to C$ and I want to write its inverse $f^{-1}$ in terms of the inverse of the sections of the function $f$.
The sections are $$a\to f(a,b) \quad (\forall b\in B,a\in A) \equiv f^b(a)$$ and $$b\to f(a,b) \quad (\forall a\in A,b\in B) \equiv f_a(b)$$
Do the inverse functions $(f^b)^{-1}$ and $(f_a)^{-1}$ exist? Because for some fixed $a$, there is not necessarily a $B_0\in B$ such that $f_a(B_0) = O$. So at best $f_a(B_0)\subset O$. If they do exist how are they defined? If not, what is the closest concept?