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Suppose we have an abelian category $\mathcal{A}$ and a topological space $X$. Let $\mathscr{F}^{\mathcal{A}}$ be the category of all sheaves of $\mathcal{A}$ objects over $X$. Is $\mathscr{F}^{\mathcal{A}}$ guaranteed to be an abelian category? I don't really have any evidence that this should be true other than that it happens to be true when $\mathcal{A}$ is Ab or $R$-Mod, and those are the principle abelian categories.

Of course, by the Freyd–Mitchell embedding theorem $\mathcal{A}$ is always equivalent to a full subcategory of $R$-Mod, but I'm not familiar enough with sheaves or the embedding theorem itself to know if the "subcategory" factor could play a problem here.

In the event that the answer to the posed question is "no", is there some extended criterion for $\mathscr{F}^{\mathcal{A}}$ to be abelian?

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It seems that the answer to the posed question is "yes".

It is noted, but without proof (there might be no difficulty with a direct proof without using the Freyd-Mitchell embedding theorem or whatever non-trivial result), in Weibel An introduction to homological algebra (1.6 "More on abelian categories").

The category Sheaves(X) of sheaves on X with values in any abelian category is an abelian category, however it is not an abelian subcategory of Presheaves(X).

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