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Allow me to generalise to $\kappa$-accessible categories for infinite regular cardinals $\kappa$. Your guess for (2) is correct: if $F$ preserves $\kappa$-small colimits then $\tilde{F}$ preserves colimits. The proof is a little bit indirect. Proposition. Let $\mathcal{I}$ be a category and let $\mathcal{C}$ be a small category with $\kappa$-small colimits. ...


Since this result is derived by Lieberman, Rosicky, and Vasey as a corollary of some more sophisticated constructions with more sophisticated goals, I think it might be worth "compiling out" the proof here. It turns out to be not so bad. Let $\mathcal C$ be a locally $\lambda$-presentable category, and recall the following fact: For any $C \in \...


The last Remark in my joint paper gives a positive answer to the Question.

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