Consider the presheaf category $\left[C, \mathbf{Set}\right]$ where $C$ is small. I have read that this is a locally finitely presentable category. This makes sense to me except one detail:
Why is the full subcategory on all finitely presentable presheaves essentially small?
I can't come up with a set-theoretic justification for this. A proof sketch would be greatly appreciated.
There are certainly small many (co?)representable functors. So there is also only (essentially) small many presheaves that are finite colimits of representable functors. Now, every presheaf $F$ is a colimit of representable functors, and by modifying the underlying diagram of the colimit, one can show that $F$ is a filtered colimit of finite colimits of representable functors: the idea is to consider "colimits of all finite subdiagrams" first and then assembling them into a filtered system by "ordering them by inclusion" (the process is described precisely here in E. Wofsey's answer, only for limits instead of colimits).
Now suppose that $F$ is itself finitely presentable, and consider such filtered colimit of finite colimits of representables as described above: $F \stackrel{\simeq}\rightarrow \varinjlim_i G_i$. Since $F$ is finitely presentable, this isomorphism factors through one of the $G_i$'s, resulting in an (componentwise) injection $F \hookrightarrow G_i$.
Thus, all finitely presentable presheaves are, up to isomorphism, sub-presheaves of finite colimits of representable presheaves, and there are certainly only small many of those.