In category theory, does $X \leq Y$ have a nice characterization in terms of the existence of a morphism $X \rightarrow Y$? Given objects $X$ and $Y$ of a category, write $X \leq Y$ iff for all objects $A$ and $B$ and all arrows $f : A \rightarrow B$, we have that if $f$ factors through $X$, then $f$ factors through $Y$. Thus the statement $X \leq Y$ is somehow expressing the idea that $X$ is "smaller" or "more narrow" than $Y$.
Question. Does the statement $X \leq Y$ have a nice characterization in terms of the existence of an arrow $X \rightarrow Y$ having a certain property?
(Probably there is an obvious answer, but I'm not very good at category theory, so everyone's help is very much appreciated.)

The situation in $\mathbf{Set}$ turns out to be generic: in any category, $X \le Y$ if and only if there exists a split monomorphism $X \to Y$, or equivalently, if and only if there exists a split epimorphism $Y \to X$.
Indeed, the morphism $\mathrm{id} : X \to X$ factors through $X$, so if $X \le Y$, then there must exist $s : X \to Y$ and $r : Y \to X$ such that $r \circ s = \mathrm{id}_X$; conversely, given such $s : X \to Y$ and $r : Y \to X$, for any $f : A \to X$ and $g : X \to B$, we have $g \circ f = (g \circ r) \circ (s \circ f)$, so indeed $X \le Y$.
