Actually, thinking about it, we could give a dual presentation by restricting \$$\mathbf{Feas}\$$.

Then restricting the posets in \$$\mathbf{Feas}\$$ to only sets gives the category \$$\mathbf{Rel}\$$ (as I noted above), restricting thre feasiblity relations in \$$\mathbf{Feas}\$$ to only monotone functions gives the category \$$\mathbf{Poset}\$$, and restricting in both way gives the category \$$\mathbf{Set}\$$.

If there was a category \$$\mathbf{Porel}\$$ of posets as objects and relations as morphisms, then \$$\mathbf{Feas} \cong \mathbf{Porel}\$$ .