There is a forgetful functor $U:\mathbf{Cat} \to \mathbf{Graph}$, which assigns a (small) category to its underlying (small) graph. Also, it has a left adjoint $F:\mathbf{Graph} \to \mathbf{Cat}$, called free functor, which assigns a graph to the freely-generated category.
My question is: is there a right adjoint (cofree) to $U$? If exists, how to construct? if not, why not?
I guess this is not as easy as the case $\mathbf{Top} \to \mathbf{Set}$ or $\mathbf{Cat} \to \mathbf{Set}$ because $U$ preserves coproducts and because a graph has edges and vertices.
Thanks.