For any functor \(\mathbf{1} \xrightarrow{F} \mathbf{Set}\) one can extract a set, \(F(1)\). Show that for any set \(S\) there is a functor \(\mathbf{1}\xrightarrow{F_S}\mathbf{Set}\) such that \(F_S(1)=S\)
Define \(F_S\) to send the object of 1 to \(S\) and preserve identity morphisms. There is no nontrivial composition to enforce, so this is a valid functor.