There are various equivalent definitions of the concept of a category, but it seems to me that in all definitions the class $hom(\mathcal C)$ of morphisms of a category $\mathcal C$ is the disjoint union of the morphism sets $hom(A,B)$, where $A, B$ are the objects of $\mathcal C$.
The "classic" definition requires that we given for any two objects $A, B \in ob(\mathcal C)$ a set of morphisms $hom(A, B)$ such that
$$hom(A, B) \cap hom(A', B') = \emptyset \text{ if } (A, B) \ne (A', B') .$$ The class of all morphisms of $\mathcal C$ is then $$hom(\mathcal C) = \bigcup_{A, B \in ob(\mathcal C)} hom(A, B) .$$
An alternative definition takes the class $hom(\mathcal C)$ of all morphisms of $\mathcal C$ as a basic structural component of $\mathcal C$ plus two functions $dom, cod : hom(\mathcal C) \to ob(\mathcal C)$ which associate to a morphisms $f$ its domain $dom(f) $ and its codomain $cod(f)$, thus partitioning $hom(\mathcal C)$ into disjont morphism sets between the objects.
It is obvious that these definitions are equivalent, but to prove it we need the disjointness of the morphism sets $hom(A,B)$. This shows that disjointness is a very natural and useful requirement.
Nevertheless we could omit it in the "classic definition".
My question: Is there a deeper reason for the disjointness requirement, or is it just convenience?
Here are some arguments pro skipping disjointness.
If necessary, we can replace $\mathcal C$ by the category $\mathcal C'$ which has the same objects and $$hom'(A,B) = \lbrace (A, f,B) \mid f \in hom(A,B) \rbrace .$$ This gives pairwise disjoint morphisms sets, though it is just a "formal trick".
Frequently categories are defined in a way not giving pairwise disjoint morphism sets.
For example, in
Adámek, Jiří, Horst Herrlich, and George Strecker. Abstract and concrete categories. Wiley-Interscience, 1990.
one finds the following constructions:
Rel with objects all pairs $(X, \rho)$, where $X$ is a set and $\rho$ is a (binary) relation on $X$. Morphisms $f : (X, \rho) \to (Y, \sigma)$ are relation-preserving maps; i.e., maps $f : X \to Y$ such that $x \rho x'$ implies $f(x) \sigma f(x')$.
Pointed Categories
(a) Show that there is a category whose objects are all pairs of the form $(A, a)$, where $A$ is a set and $a \in A$ and $$hom((A, a), (B, b)) = \lbrace f | f : A \to B \text{ and } f(a) = b \rbrace .$$
