Essential smallness of finitely presentable subcategory 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.
 A: 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. 
