Let \(z \in P\) be an element of a preorder, and consider the corresponding category \(\mathcal{P}\). Show that z is a terminal object iff it is a top element in \(P\), i.e. \(\forall c \in P: c \leq z\)
There is a morphism from every object if every object is less than z, and the uniqueness comes from the fact that preorders are thin categories.