Consider the full subcategory of Top consisting of all spaces $X$ such that a subset $A$ of $X$ is closed if and only if $A \cap K$ is closed in $K$ for all subspaces $K$ of $X$ which are countably compact, that is every open covering has a countable subcovering. It would be convenient for a proof I'm trying to write if this were known to be cartesian closed; is it known whether this is the case?