$\quad$\n1. An object $c$ in a category $\mathsf{C}$ is initial if and only if the functor $\mathsf{C}(c,-) \colon \mathsf{C} \to \textup{\textsf{cat}}$ is naturally isomorphic to the constant functor $* \colon \mathsf{C} \to \textup{\textsf{cat}}$ that sends every object to the singleton set.\n2. An object $c \in \mathsf{C}$ is terminal if and only if the functor $\mathsf{C}(-,c) \colon \mathsf{C}^\mathrm{op} \to \textup{\textsf{cat}}$ is naturally isomorphic to the constant functor $* \colon \mathsf{C}^\mathrm{op} \to \textup{\textsf{cat}}$ that sends every object to the singleton set.