What are the split monomorphisms in $\mathbf{Cat}$, forgetting 2-categorical structure?
So far I know that they are faithful and injective on objects, since those functors are the monomorphisms in $\mathbf{Cat}$. The context in which this question came up makes me guess that fullness might play a role or that the image is closed under taking isomorphism classes. I have no idea how to prove that though.
I don't know anything interesting to say about this condition; it doesn't strike me as very natural, among other things because it ignores natural transformations.
The natural condition, as Kevin Carlson says, is a functor being split by an adjoint. Formally, if $F : C \to D$ is a left adjoint and $G : D \to C$ is its right adjoint, we have unit and counit natural transformations
$$\varepsilon : \text{id}_C \to GF$$ $$\eta : FG \to \text{id}_D$$
and it's an interesting and natural question to ask what happens if either of these is an isomorphism. It turns out (this is a straightforward exercise) that the unit $\varepsilon$ is an isomorphism iff $F$ is fully faithful and the counit $\eta$ is an isomorphism iff $G$ is fully faithful.
Related and also quite interesting is the notion of an idempotent adjunction.