Presheaf category is locally finitely presentable Let $S$ be a small category. Consider its presheaf category $\widehat{S} = [S^{\mathrm{op}},\mathsf{Set}]$. Is there a direct way to see that $\widehat{S}_{\mathrm{fp}}$ is essentially small, and that every object of $\widehat{S}$ is a filtered colimit of objects in $\widehat{S}_{\mathrm{fp}}$? This is true by the various characterizations of locally finitely presentable categories, but I wonder if there is a more direct way. I suspect that $\widehat{S}_{\mathrm{fp}}$ consists of those presheaves which are finite colimits of representable presheaves.
 A: Exercise: If you have finite coproducts and filtered colimits, then you have all coproducts.
Very standard result 1: If you have coequalizers and arbitrary coproducts, then you have all colimits.
Very standard result 2: Presheaves are colimits of representables.
Let $\mathcal{I}$ be a filtered category and $D : \mathcal{I}\to[S^{op},\mathbf{Set}]$. Similarly, let $\mathcal{J}$ be a finite category (i.e. a finite number of arrows) and $D' : \mathcal{J}\to[S^{op},\mathbf{Set}]$. I'll use (co)end syntax for (co)limits because it provides a nice binding syntax, but I'm not actually using any (proper) (co)ends.


*

*Representables are compact: $$\begin{align}
\mathsf{Nat}\left(\mathsf{Hom}(-,X),\int^{I\in\mathcal{I}}D(I)\right)
& \cong \left(\int^{I\in\mathcal{I}}D(I)\right)\!(X) \\
& \cong\int^{I\in\mathcal{I}}D(I)(X) \\
& \cong\int^{I\in\mathcal{I}}\mathsf{Nat}(\mathsf{Hom}(-,X),D(I))
\end{align}$$

*Finite colimits of compact objects are compact: $$\begin{align}
\mathsf{Nat}(\int^{J\in\mathcal{J}}D'(J),\int^{I\in\mathcal{I}}D(I))
& \cong \int_{J\in\mathcal{J}}\mathsf{Nat}\left(D'(J),\int^{I\in\mathcal{I}}D(I)\right) \\
& \cong \int_{J\in\mathcal{J}}\int^{I\in\mathcal{I}}\mathsf{Nat}(D'(J),D(I)) \\
& \cong \int^{I\in\mathcal{I}}\int_{J\in\mathcal{J}}\mathsf{Nat}(D'(J),D(I)) \\
& \cong \int^{I\in\mathcal{I}}\mathsf{Nat}(\int^{J\in\mathcal{J}}D'(J),D(I)) \\
\end{align}$$


Thus $\widehat{S}_{fp}$ contains at least the representables and finite colimits of them, and thus coequalizers and finite coproducts. We can get arbitrary colimits, and thus all presheaves, via a filtered colimit of finite coproducts and coequalizers.
