Has the covariant Hom-functor of the category of additive categories a left adjoint?

Question

Let $\mathsf{Add}$ denote the (strict) 2-category of small additive categories and additive functors. Because categories of additive functors are itself additive, we have for each additive category $\mathcal{A}$ a covariant Hom-functor

$$\mathsf{Add}(\mathcal{A},-) : \mathsf{Add} \to \mathsf{Add} $$

in a 2-categorical sense, i.e. for each additive functor $F:\mathcal{B}\to\mathcal{C}$ we obtain an additive functor

$$F_*^\mathcal{A} : \mathsf{Add}(\mathcal{A},\mathcal{B}) \to \mathsf{Add}(\mathcal{A},\mathcal{C})$$

Does the functor $\mathsf{Add}(\mathcal{A},-)$ have a left adjoint in a 2-categorical sense? Does there exists for each additive category $\mathcal{B}$ an additive category "$\mathcal{B}\otimes\mathcal{A}$" such that we have for each additive category $\mathcal{C}$ a natural isomorphism (or at least an equivalence) of categories

$$\mathsf{Add}(\mathcal{B} \otimes \mathcal{A},\mathcal{C}) \cong \mathsf{Add}(\mathcal{B},\mathsf{Add}(\mathcal{A},\mathcal{C}))\quad ?$$

Answer 1

Yes, additive categories are enriched in the category of abelian groups (or commutative monoids), which is a (co)complete monoidal closed category. I recommend you look at the construction of the hom-V-category and the tensor V-category in chapter 2 of Kelly’s Basic Concepts of Enriched Category Theory.