It seems to me that arrows are given with two maps $dom, codom$ 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, I think. 

[1] Jean-Pierre Serre (1977), Trees, Springer.