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).