Two short questions:

- Is there any work classifying the lattice of subcategories of an arbitrary (sufficiently small) category $\mathcal{C}$, similar to the way that the set of subsets of set $\mathcal{S}$ is isomorphic to the functions $\mathcal{S}\to\mathbf{2}$, where $\mathbf{2}$ is a two point set?

- Is there standard notation denoting the lattice of subcategories of some category?

The definitions found in [nLab][1] are phrased in terms of functors going into $\mathcal{C}$, but the definition for sets talks about functions out of the set $\mathcal{S}$. Why are things done differently?


  [1]: http://ncatlab.org/nlab/show/subcategory