In the category of (finite) simple graphs with graph homomorphisms $\mathsf{SimpGph}$, (how) can the complete graphs $K_n$ be characterized by genuinely categorical means? Are they somehow "distinguished" categorically?
Miscellaneous findings so far:
$\mathsf{SimpGph}$ has only the null graph $K_0$ as its initial object (from which one cannot construct other graphs) and no terminal object at all.
When loops are allowed the single-vertex-with-one-loop-graph is terminal, but it is not the complete graph $K_1$.
The digraph $\circ\!\!\rightarrow\!\!\circ$ of two vertices with a single arrow between them (something like the directed version of $K_2$) gives rise to a functor category which is essentially the category of all digraphs.