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 colimit, 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 $\mathop{colim}(D)$ 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?

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?