my question is related to the tensor product in $k$-linear additive categories. If you have such a category, an object $A$ and a finite dimensional $k$-vector space $V$, then one knows that the tensor product $V\otimes_kA$ exists.
One usually chooses a basis of $V$ and sets $A\otimes_kV$ simply as $A^n$ and verifies the universal property
$Hom(V\otimes_k A,B)\simeq Hom_k(V,Hom(A,B))$ directly.
Now it is not clear for me inasfar this construction is canonical, as you chose a basis of $V$.
And if $V$ is simply $Hom(A,B)$, then you have a morphism
$Hom(Hom(A,B)\otimes A,B)$ corresponding to the identity. How canonical is also this one?
Thank you very much!