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Is is true that a functor preserving both all finite colimits and all filtered colimits, preserves in fact all colimits? I read this somewhere and have tried to find a proof of it, but I can't find one.

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It is true. I don't really know any references for this, but it's not really hard to show : remember that a functor preserve colimits if it preserves coproducts and coequalizers. Now coequalizers are finite colimits, so it's enough to prove that a functor that preserves finite colimits and filtered colimits preserves coproducts; and this just follows from the fact that the coproduct of a family of objects $(X_i)_{i\in I}$ is the colimit of the coproducts over finite subsets of $I$, i.e. : $$\coprod_{i\in I} X_i=\underset{J\subset I;\\ |J|<\infty}{\operatorname{colim}} \coprod_{j\in J}X_j,$$ and this is a filtered colimit of finite colimits, as the set of finite subsets of $I$ is directed in the poset of subsets of $I$.

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