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: $F\_3 \circ U\_2$.



When are subcategories of continuous functors reflective?
When $D$ has copowers, the reflectivity of continuous functors can be seen as a case of the **orthogonal subcategory problem**. This problem asks whether the full subcategory $\mathcal{X}\_\Gamma$ of a category $\mathcal{X}$ on the objects orthogonal to a set of morphisms $\Gamma$ is reflective. There are many hypotheses that ensure a positive answer, a famous one being the condition that $\mathcal{X}$ is a locally presentable category and $\Gamma$ is a small set.


The problem so reduces because we can show that a functor $G : C \to D$ preserves the limit of $F : J\_0 \to C$ if and only if it is orthogonal in the functor category $[C,D]$ to the morphisms 
$$\text{colim}\_j C(Fj,-)\cdot d \longrightarrow C(\text{lim}\_j Fj,-)\cdot d$$
for each $d\in D$. Letting $\Gamma$ be the set of these morphisms for all $F \in J$, we have therefore that the full subcategory $[C,D]\_J$ of the $J$-continuous functors is the orthogonal subcategory $[C,D]\_\Gamma$.


Hence (keeping your assumptions of smallness) the full subcategory $[C,D]\_J$ is reflective in $[C,D]$ if $D$ is a **locally presentable category** and $J$ is a **small set**.


The classic paper on this subject is 



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> Freyd, P. J.; Kelly, G. M. Categories of continuous functors. I. J. Pure Appl. Algebra 2 (1972), 169--191. 
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where more general hypotheses are considered,
and see 



> 
> Kelly, G. M. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bull. Austral. Math. Soc. 22 (1980), no. 1, 1--83.
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> 
> 


(in particular section 12) for a more complete and cleaner treatment.


