With a category $\mathcal{C}$, it seems to me that arrows are given with two maps $dom, codom$ from the class $Ar(\mathcal{C})$ to the class $Ob(\mathcal{C})$ so that if $f\in Hom_C(X,Y)$ is given we have $dom(f)=X,codom(f)=Y$. Now, if $$f\in{\bf Hom}_\mathcal{C}(Y,Z)\cap{\bf Hom}_\mathcal{C}(Y',Z')$$ and $$g\in{\bf Hom}_\mathcal{C}(X,Y)\cap{\bf Hom}_\mathcal{C}(X',Y')$$ we must have $X=X',\ Y=Y',\ Z=Z'$. The situation is very much like in graph theory, for the definition of a Directed Multigraph, a Quiver or an Automaton. I pick the definition in [1].
A Graph (understand Directed Multigraph) $\Gamma$ consists of a set $X=vert\Gamma$ and a set $Y=edge\Gamma$ and two maps $o:\ Y\to X$ (origin) and $t:\ Y\to X$ (target).
Late edit The MO legitimately interprets these ``functions'' as "constructed" whereas they are "thought" as univalued (it is explicit in Serre) in the mind of many category theorists.
This issue is explicitly unearthed and treated in [2] page 11 Remark 1.1.2 (e)
``If $f ∈ \mathcal{A}(A, B)$, we call $A$ the domain and $B$ the codomain of $f$. Every map in every category has a definite domain and a definite codomain. (If you believe it makes sense to form the intersection of an arbitrary pair of abstract sets, you should add to the definition of category the condition that $\mathcal{A}(A, B) \cap \mathcal{A}(A', B') = \emptyset$ unless $A = A'$ and $B = B'$.)''
[1] Jean-Pierre Serre (1977), Trees, Springer.
[2] Tom Leinster, Basic Category Theory, Cambridge University Press (2014)