TITLE: About the definition of category?
QUESTION [7 upvotes]: In some books the definition of a category includes the following condition:

If $(X, Y)$ and $(X^\prime, Y^\prime)$ are distinct pairs of objects then $\textrm{Hom}(X, Y)\cap \textrm{Hom}(X^\prime, Y^\prime)=\emptyset$.

Can anyone explain me the need of this? Why some authors don't include this condition?

REPLY [12 votes]: It is necessary to demand different hom sets are disjoint since otherwise a single morphism may have different domain and/or codomain, and that may affect whether that morphism is epic, monic, etc. To understand the importance of this just remember why it is important (e.g., in the category of sets and functions) to demand that the two functions $f(x)=\sin x$ where $f\colon \mathbb R \to \mathbb R$ and $g(x)=\sin x$ where $g\colon \mathbb R \to [-1,1]$ be considered different. After all, $f$ is not surjective but $g$ is. If we did not distinguish between these two functions, then is $f=g$ surjective or not?
I believe all authors include this axiom in one guise or another. All it says is that every morphism has a unique domain and a unique codomain.