I'm not sure if the following technicality should just be ignored, or if I should just learn more category theory (of which I know very little), but here it goes anyway.
Let $\mathcal{C}$ be the category whose objects are sets and whose morphisms are partial functions between sets. It is my understanding that that a proper partial bijection $f:X \to Y$ is technically not an isomorphism between its source and target (it would have to be a total function to qualify as an isomorphism). Therefore writing $f^{-1}$ to stand for the proper partial bijection $Y \to X$ that we would intuitively call the "inverse" $$\{(y,x):(x,y) \in f\}$$ of $f$ conflicts with categorical terminology as it suggests $f$ is an isomorphism and invertible in $\mathcal{C}$, which is technically untrue.
Is there special notation that is used to denote the intuitive "inverse" of a proper partial bijection $f$, or maybe a special property of morphisms that encodes this weaker sense of "invertibility"? At the very least, do most people write $f^{-1}$ anyway, despite the technicality that $f$ is not an isomorphism in $\mathcal{C}$.
You are asking for notation for the inverse relation of the relation $f$ (which happens to be a partial function).
Wikipedia lists the following: $f^{-1}, f^C, f^T, f^\sim, f^\circ$ etc. The book "Categories, Allegories" uses $f^\circ$ for example.
Also consider that the category with sets as objects and relations as morphisms can be made into a dagger category by equipping it with "taking the inverse relation", so $f^\dagger$ is also reasonable notation.
Take your pick (though I personally wouldn't recommend $f^{-1}$ for the reason you already gave).