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.
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$\begingroup$ what do you mean here by the extension of $F$ to $pro(C)$ the procompletion of $C$?? $\mathbf{Set}$ has cofiltered limits, so any functor $G \colon C \to \mathbf{Set}$ extends to the procompletion (this is a free completion) regardless of $F$ being left exact or not. $\endgroup$ – Eduardo Pareja Tobes Jul 22 '11 at 16:35

$\begingroup$ Under my assumptions F is representable on pro(C). I want to know whether this representable functor on pro(C) preserves finite colimits if F did on C. $\endgroup$ – Benjamin Steinberg Jul 22 '11 at 17:37

$\begingroup$ Ok, I see what you mean; just a little clarification: by "this representable functor on $pro(C)$" you mean 1. $pro(C)(,F)\colon pro(C)^{op} \to \mathbf{Set}$ or 2. $pro(C)(F,)\colon pro(C) \to \mathbf{Set}$?? $\endgroup$ – Eduardo Pareja Tobes Jul 24 '11 at 19:13

$\begingroup$ The covariant representable, i.e., option 2. $\endgroup$ – Benjamin Steinberg Jul 25 '11 at 18:27
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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.

$\begingroup$ No guarantees but it makes some good points. Reindexing lemmas are very important in procategory theory in order to prove structural lifting results. $\endgroup$ – Tim Porter Dec 18 '11 at 15:07