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Let $\mathbf C$ be a category with finite limits. Then a left exact functor $F\colon \mathbf C\to \mathbf{Set}$ is pro-representable and hence extends to the pro-completion $\mathbf C$. 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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what do you mean here by the extension of $F$ to $pro(C)$ the pro-completion of $C$?? $\mathbf{Set}$ has cofiltered limits, so any functor $G \colon C \to \mathbf{Set}$ extends to the pro-completion (this is a free completion) regardless of $F$ being left exact or not. – Eduardo Pareja Tobes Jul 22 '11 at 16:35
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. – Benjamin Steinberg Jul 22 '11 at 17:37
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. $pro(C)(F,-)\colon pro(C) \to \mathbf{Set}$?? – Eduardo Pareja Tobes Jul 24 '11 at 19:13
The covariant representable, i.e., option 2. – Benjamin Steinberg Jul 25 '11 at 18:27

I am not certain if he provides the answer to your question, but Dan Isaksen's paper: Calculating limits and colimits in pro-categories, Fundamenta Mathematicae 175 (2002) 175--194. is relevant I think. The point is that the reindexing lemmas in procategories provide a powerful tool for calculating limits and colimits.

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Tim, thanks. I will take a look. – Benjamin Steinberg Dec 18 '11 at 13:29
No guarantees but it makes some good points. Reindexing lemmas are very important in pro-category theory in order to prove structural lifting results. – Tim Porter Dec 18 '11 at 15:07

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