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A well known result states that, if $F:C \rightarrow D$ is a covariant functor between categories which admit finite projective limits, then $F$ is left exact if and only if it preserves finite projective limits.

I need to use this result, but unfortunately I was unable to find a reference or to prove it by myself. I would like to have one of the two.

Note: For completeness it's useful to say that the same result holds for right exact functor and finite direct limits. And that a functor preserves finite projective limits if and only if it preserves final objects and fiber products, or if and only if it preserves final objects, products and equalizers.

Edit: The definition of left exactness I suppose given is the one that can be found in wikipedia: http://en.wikipedia.org/wiki/Exact_functor. Which is: "$F$ is left exact if it brings the short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ of objects (and morphisms) of $C$ to an exact sequence $0 \rightarrow F(A) \rightarrow F(B) \rightarrow F(C) \rightarrow ...$ of objects of $D$". I don't think the result is untrue since also the page I linked states the result.

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    @Fallen: Thank you very much, that makes a lot of sense. I admit that I have never thought about these things this way but it seems to be an extremely good point. Also, I'm probably a bit spoiled by the fact that most of the categories I care about usually are finitely bicomplete.2011-05-14

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