Right adjoint of the inclusion of preorders into small categories

Question

Let $\mathrm{Pre}$ denote the category of preorders, and $\mathrm{Cat}$ the category of small categories. Since every preorder is a category, and monotone map of preorders is a functor, we have the obvious inclusion functor $I\colon\mathrm{Pre}\to\mathrm{Cat}$. The functor $I$ has a left adjoint $F\colon\mathrm{Cat}\to\mathrm{Pre}$ (which simply "glues together" all arrows with the same domain and codomain). But does it have a right adjoint?

Answer 1

Consider the embedding $i$ of the discrete two object category $1+1$ to the category $2$ of two objects and a single nonidentity arrow from one to the other one.

The pushout of $i$ and $i$ is $2$ (along the identity maps) in $Pre$, but it contains two parallel arrows in $Cat$, so the inclusion functor doesn't preserve all colimits, hence can't have a right adjoint.