Fix a field $k$ and suppose $\mathcal{C}$ and $\mathcal{D}$ are $k$-linear additive categories and are enriched over the category $\mathcal{V}$ of finite-dimensional $k$-vector spaces. So we have copower functors
$$
 \mathcal{C} \times \mathcal{V} \to \mathcal{C} \quad \text{and} \quad \mathcal{D} \times \mathcal{V} \to \mathcal{D}.
$$
Now suppose $F \colon \mathcal{C} \to \mathcal{D}$ is a $k$-linear functor. Does it follow that the diagram
$\require{AMScd}$
\begin{CD}
 \mathcal{C} \times \mathcal{V} @>>> \mathcal{C} \\
 @V F \times \text{id}\_{\mathcal{V}} VV & @VV F V \\
 \mathcal{D} \times \mathcal{V} @>>> \mathcal{D}
\end{CD}
commutes (where the horizontal arrows are the copower functors)? If not, are there some natural assumptions on $F$ that make this diagram commute? I have a particular example in mind, where $\mathcal{C}$ and $\mathcal{D}$ are categories of modules over finite-dimensional $k$-algebras and $F$ is a functor given by tensoring with a bimodule. So I'd be happy with assumptions that are natural in that setting.



Do copowers commute with k-linear functors?
Yes, this is true (up to natural isomorphism). The simplest way to see this is just to write down what the copowers are explicitly. Let us take a skeleton of $\mathcal{V}$ consisting of all vector spaces of the form $k^n$, and write $\otimes$ for copowers. Then on objects, $A\otimes k^n$ is just a direct sum of $n$ copies of $A$. And on morphisms, given $f:A\to B$ and $T:k^n\to k^m$, the induced map $f\otimes T:A\otimes k^n\to B\otimes k^m$ can be described as follows. The domain is $A^n$ and the codomain is $B^m$, so we can describe $f\otimes T$ as an $n\times m$ matrix of morphisms $A\to B$. The $ij$ entry of this matrix is just $T\_{ij}f$, where $T\_{ij}\in k$ is the $ij$ matrix entry of $T$.


All of this structure depends only on the $k$-linear structure of $\mathcal{C}$, so it is preserved by any $k$-linear functor. Explicitly, morphisms $A^n\to B^m$ are in bijection with matrices of morphisms $A\to B$ by composing with the inclusion and projection maps $A\to A^n$ and $B^m\to B$, and this correspondence is preserved by any additive functor. ($k$-linearity is also needed so that you know the scalar products $T\_{ij}f$ are preserved.)


