I'm wondering about categorifications of Zorn's lemma along the following lines.

Lemma: if $\mathbf{C}$ is a small category in which every directed diagram of monomorphisms has a cocone of monomorphisms, then there is an object $A$ such that any monomorphism $A \rightarrowtail B$ splits.

*Proofsketch*: If there were no such object, we could use the axiom of choice to define a function $f$ that assigns to each directed diagram $D$ of monomorphisms, a monomorphism with domain $\mathop{cocone}(D)$ that does not split. Define an ordinal-indexed diagram $D$ by setting $D(\alpha) = f\big( D(\beta) \mid \beta<\alpha \big)$. Because $D$ consists of monomorphisms that don't split, it cannot contain any cycles. But this contradicts smallness of $\mathbf{C}$.

Can this be generalised? (E.g. do we need smallness and monics?) Are other versions known? Are there similar categorical existence statements that are provable without the axiom of choice?