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Does the tensor product of abelian categories commute with pullbacks? In more detail:

Let $k$ be a field. We consider $k$-linear small abelian categories $\mathcal{B},\mathcal{A}_0,\mathcal{A}_1,\mathcal{A}_2$. Assume that $\mathcal{A}_2 \to \mathcal{A}_0 \leftarrow \mathcal{A}_1$ are exact $k$-linear functors. The pullback $\mathcal{A}_2 \times_{\mathcal{A}_0} \mathcal{A}_1$ has objects $(a_1,a_2,\sigma)$, where $a_i \in \mathcal{A}_i$ and $\sigma$ is an isomorphism between the images in $\mathcal{A}_0$. See arXiv:1212.1545 for the definition of $\boxtimes_k$, the tensor product of finitely cocomplete $k$-linear categories. The universal properties yield a right exact $k$-linear functor $$(\mathcal{A}_1 \times_{\mathcal{A}_0} \mathcal{A}_2) \boxtimes_k \mathcal{B} \to (\mathcal{A}_1 \boxtimes_k \mathcal{B}) \times_{(\mathcal{A}_0 \boxtimes_k \mathcal{B})}(\mathcal{A}_2 \boxtimes_k \mathcal{B}).$$

Question. Is this functor an equivalence of categories?

I hope that the answer is yes, but the problem is that I don't know an efficient description of the objects and the morphisms in the tensor product. The categories are not assumed to be of finite length or rigid. I'm mainly interested in categories of coherent sheaves. If necessary we may assume that $\mathcal{A}_0$ and $\mathcal{A}_2$ are categories of f.p. modules over commutative f.g. $k$-algebras. But $\mathcal{B}$ stays arbitrary (if $\mathcal{B}$ is such a module category, the answer is clearly yes).

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By theorem 7 in arXiv:1212.1545 the tensor product is a "nice reflection" of the ordinary tensor product into the finitely cocomplete categories. Have you tried to obtain the needed equivalence using this universal property and the fact that the ordinary tensor product commutes with pullbacks (?), and some sort of nice interaction of pullbacks and the ordinary tensor product with the right exactness (the "niceness" of the "reflection" being a certain right exactness property)? –  Dimitri Chikhladze Jul 9 at 0:39
    
Sorry for a vague language, I hope the idea is understandable :) –  Dimitri Chikhladze Jul 9 at 0:39
    
Sorry but this doesn't help me. I doubt that universal properties will help (we interchange colimits with limits). Also it's unclear to me what is nice here. The transfinite recursion in the reflection is horrible. –  Martin Brandenburg Jul 17 at 20:13
    
You are right it's an interplay between limits and colimits. I didn't mean it was a trivial check of some universal property. All I meant is maybe one can get away without an explicit description of the tensor product. –  Dimitri Chikhladze Jul 17 at 21:04
    
The cocomplete product works this way. First we're taking an ordinary tensor product of k-linear categories and then universally embed it into a cocomplete category. By an embedding is meant a bifunctor which is exact in each variable. This is what I mean by being nice. –  Dimitri Chikhladze Jul 17 at 21:11

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