This question is probably stupid and definitely bureaucratic, but

>Is writing $f\circ g$ for the composition of morphisms in the 'many hom-classes' definition of a category unambiguous?

The many hom-classes definition of a category (as given e.g. on the [nlab](https://ncatlab.org/nlab/show/category)) says that for each pair of arrows 
$(f,g)\in{\bf Hom}_\mathcal{C}(Y,Z)\times{\bf Hom}_\mathcal{C}(X,Y)$ we 'have an arrow' $f\circ g\in{\bf Hom}_\mathcal{C}(X,Z)$, but if the hom-classes may not be disjoint how do we know that the arrows we 'have' from identical composable arrow pairings in differing hom-class pairs match up?

Rephrased using the language of composition functions, the above definition is the same as a collection of functions $$\{\circ_{XYZ}:{\bf Hom}_\mathcal{C}(Y,Z)\times{\bf Hom}_\mathcal{C}(X,Y)\to{\bf Hom}_\mathcal{C}(X,Z)\}_{X,Y,Z\in{\bf Ob}_\mathcal{C}}.$$
The axioms then specify associativity and unitarity, but how do we know what we don't have objects $X,Y,Z,X',Y',Z'\in{\bf Ob}_\mathcal{C}$ and arrows $$f\in{\bf Hom}_\mathcal{C}(Y,Z)\cap{\bf Hom}_\mathcal{C}(Y',Z'),$$ $$g\in{\bf Hom}_\mathcal{C}(X,Y)\cap{\bf Hom}_\mathcal{C}(X',Y'),$$ so $$(f,g)\in\big({\bf Hom}_\mathcal{C}(Y,Z)\times{\bf Hom}_\mathcal{C}(X,Y)\big)\cap\big({\bf Hom}_\mathcal{C}(Y',Z')\times{\bf Hom}_\mathcal{C}(X',Y')\big),$$ such that $$\circ_{XYZ}(f,g)\neq\circ_{X'Y'Z'}(f,g)?$$ If we do the notation $f\circ g$ is obviously ambiguous -- also obvious is that we could fix this with an additional axiom (scheme?) if it were a problem.

>Is this situation already precluded by the other axioms/data present in the many hom-classes definition of a category?

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For an example where we have identical arrow pairings in differing hom-classes but composition still matches up, consider any two composable relations $R,S\in{\bf Ob_{Rel}}$. By definition $R$ and $S$ are subsets of some Cartesian squares $Y\times Z$ and $X\times Y$ (respectively), but we can take $X',Y',Z'$ to be any strict superclasses of $X,Y,Z$ (resp.) and observe that $R$ and $S$ are also subsets of $Y'\times Z'$ and $X'\times Y'$ (resp.), so $$R\in{\bf Hom_{Rel}}(Y,Z)\cap{\bf Hom_{Rel}}(Y',Z'),$$ $$S\in{\bf Hom_{Rel}}(X,Y)\cap{\bf Hom_{Rel}}(X',Y').$$ Here we obviously still have that the composition functions coincide, but this is arguably due to the fact that ${\bf Rel}$ is most naturally presented as a one hom-class category.

Again, I understand that this question is very pedantic at best -- the patience involved in any clarification is greatly appreciated.