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I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology.

My question is:

How about the sheafification functor from the category of presheaves of "sets" on $C$ to the category of sheaves of "sets" on $C$?

Is this an exact functor? (i.e. preserving finite limits and finite colimits?)

If so, how can one prove it?

In fact, I want to know whether sheafification functor preserves cartesian products or not.

Please give me any advice.

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    $\begingroup$ the answer is yes! for a reference you can use theorem 17.4.9 of kashiwara-schapira cats and sheaves. $\endgroup$
    – Yosemite Sam
    Commented Feb 26, 2012 at 10:15

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Martin has probably answered everything Hiro meant to ask, but, since "whether sheafification functor preserves cartesian products or not" didn't explicitly say finite products, let me add that sheafification will not in general preserve infinite products. Intuitively, the reason is that a section of a product of sheafifications is a family of locally defined sections of the original presheaves, and there might not be a single covering over which all those local sections are simultaneously available.

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Preservation of colimits is trivial from the adjunction, preservation of finite limits comes down - using the usual construction - to the fact that finite limits commute with filtered colimits. The latter holds in every algebraic category, in particular in (Set).

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