TITLE: Are compact objects in presheaf categories finite colimits of representables?
QUESTION [13 upvotes]: An object $x$ in a category $\mathsf{C}$ is called compact or finitely presentable if $$\mathrm{hom}(x,-) : \mathsf{C} \to \mathsf{Set}$$  preserves filtered colimits.  This concept behaves best when $\mathsf{C}$ has all filtered colimits, e.g. when it is the category of presheaves on some small category $\mathsf{X}$:
$$ \mathsf{C} = \mathsf{Set}^{\mathsf{X}^{\mathrm{op}}} $$
Every representable presheaf is compact.  In general, any finite colimit of compact objects is compact.  Thus, any finite colimit of representables is compact.
My question is about the converse: in the category of presheaves on a small category, is every compact object a finite colimit of representables?

REPLY [4 votes]: Here is another perspective on the problem, using some big guns (Gabriel-Ulmer duality). 
Given a small category $C$, let $K$ be its free finite cocompletion. This means $K^{op}$ is the free finite completion of $C^{op}$, which means in turn that for any functor $F: C^{op} \to \mathbf{Set}$, there is a finitely continuous (or left exact) functor $\tilde{F}: K^{op} \to \mathbf{Set}$ that extends $F$ along the canonical inclusion $i: C^{op} \to K^{op}$, and this extension is unique up to unique isomorphism. Put differently, restriction along $i$ induces an equivalence 
$$\mathrm{Lex}(K^{op}, \mathbf{Set}) \to \mathrm{Cat}(C^{op}, \mathbf{Set}).$$ 
In particular, the presheaf category $\mathrm{Cat}(C^{op}, \mathbf{Set})$ is locally finitely presentable. By the way, it's well known that the free finite cocompletion $K$ of a small category $C$ is simply the category of finite colimits of representables: see section 5.9 of Kelly's Basic Concepts of Enriched Category Theory.  
On the other hand, Gabriel-Ulmer duality assures us that given a locally finitely presentable category $A$, there is up to equivalence only one finitely complete category $L$ for which $A \simeq \mathrm{Lex}(L, \mathbf{Set})$. Even better, Gabriel-Ulmer duality gives a recipe for obtaining $L$: it is the dual of the category of compact objects in $A$, meaning objects $a$ such that $A(a, -): A \to \mathbf{Set}$ preserves filtered colimits. 
Putting all this together, this shows that the category of compact objects in the category of presheaves over $C$ is equivalent to the free finite cocompletion of $C$, or to the category of finite colimits of representable presheaves.