Let $\mathcal{C}$ be a category. We know that every presheaf $\mathcal{C}^{op} \to Set$ is a colimit over its slice category of representable functors. I'm wondering when every such colimit is a directed limit, and thus when the category $Ind(\mathcal{C})$ is equivalent to the category of presheaves on $\mathcal{C}$.
My idea for a sufficient condition: If $\mathcal{C}$ is small and has coproducts, then every colimit is directed. This is pretty obvious, since the coproduct of two objects provides the upper bound in the slice category of a presheaf, thus making it a directed set (it's a set by smallness of $\mathcal{C}$).
Is my argument correct that this is a sufficient condition for $Ind(\mathcal{C}) \cong PSh(\mathcal{C})$? If not, are there any other necessary and sufficient conditions for this to be the case?