A monad on a preorder $(\mathsf{P}, \leq)$ is given by an order-preserving function $T \colon \mathsf{P} \to \mathsf{P}$ that is so that $p \leq Tp$ and $T^2p \leq Tp$. If $\mathsf{P}$ is a poset, so that isomorphic objects are equal, these two conditions imply that $T^2 p = Tp$. An order-preserving function $T$ so that $p \leq Tp$ and $T^2 p = Tp$ is called a closure operator. Dually, a comonad on a poset category $(\mathsf{P}, \leq)$ defines a kernel operator: an order-preserving function $K$ so that $Kp \leq p$ and $K p = K^2 p$.