Let $\mathbf C$ be a category with finite limits. Then a left exact functor $F\colon \mathbf C\to \mathbf{Set}$ is prorepresentable and hence extends to the procompletion $\mathbf C$. My question is whether it is true that the extension of $F$ preserves finite colimits whenever $F$ does, and if so what is a reference? I'm also curious about arbitrary colimits but for the application I care about, finite colimits (or even coequalizers) are enough.

I am not certain if he provides the answer to your question, but Dan Isaksen's paper: Calculating limits and colimits in procategories, Fundamenta Mathematicae 175 (2002) 175194. is relevant I think. The point is that the reindexing lemmas in procategories provide a powerful tool for calculating limits and colimits. 

