Definition: Let $\mathcal C$ be a locally small category. An object $c \in\mathcal C$ is called $\textbf{compact}$ if $\mathrm{hom}_{\mathcal C} (c,-)$ preserves filtered colimits.
Theorem: For $\mathcal{S}\mathrm{et}$, an object is compact if and only if it is a finite set.