Suppose we have objects $X$ and $Y$, and some set of morphisms $\text{Hom}(X,Y)$.
Now suppose there is a third object $Z$, and a morphism $f:X\to Z$, such that, for each $g\in \text{Hom}(X,Y)$ there is a $g_Z\in \text{Hom}(Z,Y)$ with $g=g_Z\circ f$.
Is there a name for this pair $(Z,f)$?