Let $k$ be a field. Let $\mathcal{C},\mathcal{D}$ be finitely cocomplete $k$-linear categories, which are essentially small. Then Kelly's tensor product $\mathcal{C} \boxtimes \mathcal{D}$ is a finitely cocomplete $k$-linear category together with an universal functor from $\mathcal{C} \times \mathcal{D}$ which is right exact and $k$-linear in each variable (also denoted by $(A,B) \mapsto A \boxtimes B$). For an overview of this construction, see section 2.3 in Schäppi's paper on ind-abelian categories.

Roughly, it is constructed as a full subcategory of the category $L$ of $k$-linear functors $F : (\mathcal{C} \otimes_k \mathcal{D})^{op} \to \mathsf{Vect}_k$ with the property that for every exact sequence $A'' \to A' \to A \to 0$ in $\mathcal{C}$ and all $B \in \mathcal{D}$ the sequence $0 \to F(A,B) \to F(A',B) \to F(A'',B)$ is exact, and similarly for the other variable. Representable functors lie in $L$. The crucial step is to observe that $L$ is an orthogonal class in the category of all $k$-linear functors, hence reflective. In particular, it is cocomplete. Now $\mathcal{C} \boxtimes \mathcal{D}$ is the closure of the representable functors under finite colimits taken in $L$.

In my research I need a more explicit description of the objects in $\mathcal{C} \boxtimes D$. Unfortunately, the reflector is dreadful and isn't useful at all.

First of all, is it true that every object $M$ in $\mathcal{C} \boxtimes \mathcal{D}$ can be written as a cokernel of a map of the form $\oplus_j (A'_j \boxtimes B'_j) \to \oplus_i (A_i \boxtimes B_i)$? The answer is yes when $C,D$ are ind-abelian (Lemma 6.5 in Schäppi's paper). Secondly, and more important for me: Given that we know the objects, what are the morphisms? If $A \in \mathcal{C}, B \in \mathcal{D}$, how can we describe $\hom(A \boxtimes B,M)$ explicitly in terms of such a presentation of $M$?


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