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More generally, the Theorem of Eilenberg-Watts says the following: The category of cocontinuous functors $\mathrm{Mod}(R) \to \mathrm{Mod}(S)$ is equivalent to the category of $(R,S)$-bimodules. A bim …

If $\mathcal{A}$ is some abelian category and $S$ is some set of objects in $\mathcal{A}$, then its "abelian closure" $\overline{S}$ should be the smallest abelian full subcategory of $\mathcal{A}$ co …

Any additive functor between additive categories preserves finite coproducts and hence finite products, since they can be characterized as biproducts. A right exact functor preserves all colimits iff …

What about representable functors? $\hom(M,-) : \mathrm{Mod}(R) \to \mathrm{Ab}$ is right exact iff $M$ is projective, and it preserves direct sums iff $M$ is even finitely generated projective. Of co …

In short or long exact sequences in abelian categories the following is used very often: If $0 \to A \to B \to 0$ is exact, then $A \to B$ is an isomorphism. This is the same as saying that every morp …