Is this mathematically unambiguous. If not, what notation and vocabulary should I use? Left/right map inverses. The condition is:

... given two maps $f: A\to B,$  and $g :B \to A$, there exists no map $h : B \to A$ such that $h$ is surjective and $f = h g h^{-1}$, or $h$ is injective and $f = h^{-1} g h$ ? 

Specifically all objects $A,B$ are kleene star closures of differing finite alphabets: $A = \Sigma^*$, $A = \Sigma'^*$ and all maps $f, g, h$ are string homomorphisms.
I like the "-1" notation. Should I add "for some left/right inverse $h^{-1}$"?
 A: The notation $(-)^{-1}$ unequivocally means "inverse" in this context, i.e. a morphism such that $h \circ h^{-1} = \operatorname{id}$ and $h^{-1} \circ h = \operatorname{id}$ (when it exists, and when it does it is unique). It is not reasonable to use that notation for a mere left/right inverse.
Even if you clarify that $h^{-1}$ is just a left/right inverse (and if you absolutely want to use that notation you definitely should), people will get confused. If I read the passage you wrote, it wouldn't even cross my mind to check what $h^{-1}$ means, since this is such a standard notation; but then I would wonder why $h$ is invertible. The previous scenario is not a hypothetical: this is exactly what happened when I read your question...
Another point to consider: you use the same notation ($h^{-1}$) for two different things (left inverse and right inverse) in the same sentence. Using the same notation for two different things in the same text is already a big no-no, but in the same sentence it's particularly egregious.
Remember that you write math for others too, and if a notation is likely to confuse everyone I would just drop it and find a new one. In your case, I would just choose a letter for the right inverse and write something like:

... given two maps $f: A\to B,$  and $g :B \to A$, there neither exists a surjective map $h : B \to A$ with right inverse $k$ such that $f = h g k$, nor any injective map $h$ with left inverse $l$ such that $f = l g h$ ? 

Another name for "right inverse" is "section", so if you want to emphasize that it's a right inverse you can call it $s$. And another name for left inverse is "retraction" so you can call it $r$. But I would never call either $h^{-1}$.
