Let $J$ be a collection of small categories (to be thought of as diagrams in a category). Let $C$ be a small category with all $J$-limits (i.e. for every $J_0 \in J$ and every functor $F:J_0\rightarrow C$, $\lim F$ exists in $C$). Let $D$ be any category with all $J$-limits (such as Set).

Consider the category $[C,D]_J$ of functors from $C$ to $D$ which preserve all $J$-limits, with natural transformations as morphisms.

**Question:** What conditions on $J$ must hold in order for $[C,D]_J$ to be a reflective subcategory of $[C,D]$ (the category of all functors from $C$ to $D$)?

I ran into this question when investigating the case of two collections of diagrams $J \subseteq J'$, so that $[C,D]_{J'} \subseteq [C,D]_J$ (assuming $C$ and $D$ have all $J'$-limits). Since all these categories are full subcategories, it turns out that $[C,D]_{J'}$ is a reflective subcategory of $[C,D]_{J}$ if $[C,D]_{J'}$ is a reflective subcategory of $[C,D]$. (if $U_2$ is fully faithful, and $U_3 = U_2 \circ U_1$ has a left adjoint $F_3$, then $U_1$ has a left adjoint: $F_3 \circ U_2$.