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There is a paper by Mark Hovey which discusses incarnations of the Eilenberg-Watts theorem in homotopy theory. First he reviews the algebraic version and provides an equivalent formulation wherein the …

Indeed, you do not need the assumption about the class of subobjects being a set. The statement is true in any cocomplete abelian category, i.e., one with colimits, as you wrote (in fact, arbitrary co …

I get the sense that the OP is a relatively new user of MO and trying to learn about monoidal abelian categories. While the comment gives the core idea answering the OP's question, I wanted to point o …

Since $A$ is an abelian category, it is in particular additive, i.e. has a zero object, a product functor: $A\times A\rightarrow A$, and is Ab-enriched, i.e. hom sets are abelian groups. Since $B$ is …

In this case, there's no difference between the direct sum and the direct product, because in every degree $n$, we are only taking the direct sum of two things. With reference to the answer you linked …

I found a reference that proves some of the subresults in the question. I also left two comments below the question about places I searched that did NOT contain this result. The reference I found was …

The concept you're asking about has been studied by Christensen and Hovey in Quillen model structures for relative homological algebra. Specifically, see Example 3.1 on page 17 of the pdf. In this pap …

The literature about constructing model structures on abelian categories has grown significantly since Hovey's book came out. In particular, there is now a connection between this process and cotorsio …

The kernel and cokernel can be defined in any pointed category $\mathcal C$ with finite limits and colimits. Recall that a category is pointed it has an object that is both initial and terminal, and w …