Exponential ideal

A full subcategory $\mathcal{D}$ of a cartesian closed category $\mathcal{C}$ is an exponential ideal when $B^A$ belongs to $\mathcal{D}$ for every $B\in \mathcal{D}$ and every $A\in \mathcal{C}$.

For a reflective subcategory $\mathcal{D}$ of a cartesian closed category $\mathcal{C}$, with reflector $L$, the subcategory $\mathcal{D}$ is an exponential ideal if and only if the canonical comparisonis an isomorphism for every $A,B$. The proof repeatedly curries a map into an object of $\mathcal{D}$ and factors it through the unit of the reflection.