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 fimite 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?

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?