nLab
exponential ideal

A class $I$ of objects in a cartesian closed category $C$ is called an exponential ideal if whenever $Y\in I$ and $X\in C$, the exponential object $Y^X$ is in $I$. Of course, in particular this implies that $I$ is itself cartesian closed.

If $I$ is a reflective subcategory, then it is an exponential ideal if and only if its reflector $C\to I$ preserves finite products (see A4.3.1 in the Elephant).