Reflector product criterion for an exponential ideal
= Reflector product criterion for an exponential ideal
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 comparison
$$
L(A\times B)\longrightarrow LA\times LB
$$
is 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.