What do we lose if we allow the Hom sets not to be pairwise disjoint? In Category Theory the $hom(A, B)$ sets are generally formulated to be pairwise disjoint. We know that set theorists do not agree to this day on whether functions are also defined by their codomain or not (e.g. is $f: \mathbb R \to \mathbb R, f(x)=x^2$ different from $g: \mathbb R \to \mathbb R_0^+, f(x)=x^2$?) and on this comment we can read that (as a consequence of the above) many category theorists require functions with different codomains to be distinct objects.
My question is now: is the disjointness such an important assumption? What problems do we face (apart from some proofs that may turn out to be longer) if we relax it? Can Category Theory be formulated without it?
 A: Here I'll assume that we're only talking about small categories; this is a convenience for exposition so I can save some words talking about proper classes. If I start to encroach on a traditionally large category, we'll just say that it lives in a larger universe that lets me think of it as small. 
A first observation is that the way a category theorist likes to think about categories, as a collection of arrows with domain and codomain functions to the objects, requires that the sets of morphisms between distinct pairs of objects are disjoint, since $\hom(a,b)$ will just be the members of $\langle \mathsf{dom},\mathsf{cod}\rangle^{-1}(\{\langle a,b\rangle\})$. The usual practice is just to adjust the notion of morphism so that this works.
But we can think instead of starting with some set of potential morphisms $\mathrm{Mor}$, and thinking of a category as being based upon a function $\hom:\mathrm{Ob}(\mathcal{C})\times \mathrm{Ob}(\mathcal{C})\to \mathcal{P}(\mathrm{Mor})$. Then it's possible for two $\hom$-sets to overlap, and it almost looks like you can get into trouble defining the composition operation if we had the following scenario: $\hom(b,c)=\hom(b,c')\neq\varnothing$, but there's an $a$ such that $\hom(a,b)$ is non-empty and $\hom(a,c)\cap\hom(a,c')=\varnothing$. In this case it clearly doesn't work to think of composition as using restrictions of a function $\mathrm{Mor}\times \mathrm{Mor}\to\mathrm{Mor}$. But this doesn't tell us there's a problem with non-disjointness; it tells us that this is the wrong way to think about composition for this way of relating objects and morphisms. If this is how you're doing categories, the sensible way to define composition is by way of a family of functions $\{f_{a,b,c}\}_{a,b,c\in\mathrm{Ob}(\mathcal{C})}$ with $f_{a,b,c}:\hom(b,c)\times\hom(a,b)\to\hom(a,c)$, which works just fine.
It's easy to see that a category described in terms of the first notion of category (morphisms with domain/codomain assignments) can be translated into one using the second notion (mapping pairs of objects to subsets of morphisms). The reverse also works if we take our set of morphisms to be $\{\langle a,b,m\rangle:m\in\hom(a,b)\}$, use the first & second projections as our domain and codomain mappings, and do a little tweaking to turn our family of composition functions into a more standard function on a single domain. It's not a difficult exercise to see that up to isomorphism, these translations are mutually inverse, and establish an equivalence of categories.
