1. Suppose that I have a sifted diagram of categories $\mathcal{C}_i$, another of the same shape $\mathcal{D}_i$, and that I have a system $F_i:\mathcal{C}_i\to\mathcal{D}_i$ commuting with the morphisms in each diagram. Then I get a functor $$(\operatorname{colim} F_i):\operatorname{colim} \mathcal{C_i} \to \operatorname{colim}\mathcal{D}_i.$$ Suppose I know that each $F_i$ is fully faithful. Under what further assumptions may I conclude that $(\operatorname{colim} F_i)$ is fully faithful? For instance, if I would assume the $F_i$'s have right adjoints, so that fully faithfulness is equivalent to the unit for the adjunction being an isomorphism, does it help?

A question which seems like a de-categorified analog of this would be:

2. Fix some co-complete category $X$, and consider some sifted diagram of objects $V_i$ in $X$, another of the same shape $W_i$ in $X$. Suppose I have a system of split monomorphisms $f_i:V_i\to W_i$. Under what conditions do I know that the natural morphism, $$(\operatorname{colim} f_i): \operatorname{colim} V_i \to \operatorname{colim} W_i$$ is also a split monomorphism?

I really care about the answer to (1), but I offer (2) in my attempt to find an analogous proof one category number down, where there may be more examples/techniques/counter-examples.

Here, in motivating the attempted analogy between (1) and (2), I am trying to replace the 2-category $\operatorname{Cat}$ of categories by a 1-category $X$, and replace a fully faithful functor by a split monomorphism (I am imagining that the $F_i$'s have a right adjoint, which might behave like a splitting to make the analogy work, though it's a stretch).

I am aware of https://math.stackexchange.com/questions/1116101/colimit-preserves-monomorphisms-under-certain-conditions, where it is explained that filtered colimits preserve monomorphisms, in the 1-categorical setting. I am hoping, perhaps too optimistically, that if we move from filtered colimits to sifted colimits, we still have a chance to preserve split monomorphisms even though we (I suppose?) no longer preserve arbitrary monomorphisms. I am further hoping that if so, the argument would generalize to the 2-categorical setting.

I'd be most grateful for any hints, or references.

Disclaimer: I have simplified the above setup a bit to make a cleaner question. In case it could mislead an expert though, let me say that the version of (1) I need involves the $\mathcal{C}_i$ being Cauchy-complete, the $\mathcal{D}_i$ being presentable, and really the two colimits are taken in their respective 2-categories, in Cauchy-complete categories for the diagram of $\mathcal{C}_i$ and in presentable categories for the $\mathcal{D}_i$ (so in particular the functors appearing in each kind of colimit are suitable colimit preserving; the shape of the indexing diagram is the same, though). I assume these are technicalities and that if the question has a nice answer, I can do that kind of adaptation myself.