Say $\mathscr{A}$ is a reflective subcategory of $\mathscr{B}$, meaning the inclusion functor $i: \mathscr{A} \to \mathscr{B}$ is fully faithful and admits a left adjoint, and $\mathscr{B}$ is enriched over itself, where for sake of simplicity we assume the tensor bifunctor to be the product.


Moreover, assume $\mathscr{A}$ is closed under this tensor, and $i$ commutes with it (again, the case of products suffice). 


What can we say about a self-enrichment of $\mathscr{A}$?



Self-enrichment of reflective subcategories of self-enriched categories
The most natural question to ask is whether $\mathcal{A}$ is closed under the internal-hom of $\mathcal{B}$, i.e. that $[A,B]\in \mathcal{A}$ whenever $A,B\in\mathcal{A}$. This is an extra assumption, but it's sometimes easier to verify by reformulating it in terms of the tensor product; the resulting condition is that the reflection is a [Hopf adjunction](https://ncatlab.org/nlab/show/Hopf+adjunction).


A better-known but stronger condition than this is that $[A,B]\in \mathcal{A}$ whenever $B\in\mathcal{A}$ (but $A\in \mathcal{B}$ is arbitrary). As Zhen mentioned, in the cartesian case at least this is called being an [exponential ideal](https://ncatlab.org/nlab/show/exponential+ideal), and is equivalent to asking that the reflector be strong monoidal (preserve products).


It is, of course, technically possible that $\mathcal{A}$ is closed but that its internal-hom is not induced from $\mathcal{B}$, but it's hard to say anything in generality about when that might happen.


