For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive and transitive binary relation), $\le$, on $S$.

Now, fix a nonempty universe $\Omega$, and let $\mathsf{Dig}$ and $\mathsf{Pre}$ be the usual categories of $\Omega$-small digraphs and $\Omega$-small presets. Do we know of any interesting/non-trivial full subcategories of $\mathsf{Dig}$ and $\mathsf{Pre}$ that happen to be isomorphic to each other? If so, I would much appreciate some references.


  [1]: http://mathoverflow.net/questions/194738/isomorphic-subcategories-of-digraphs-and-presets