As to the puzzle, we can check the idea that a closure operator is a monad of a monotone map \$$T:P \to P\$$ defined in a poset \$$\(P,\leq$$\\). The unit law implies that \$$\forall p\in P: p \leq T(p)\$$, that's OK, but the multiplication law gives only \$$\forall p \in P: T.T(p) \leq T(p)\$$, while we need equality \$$T.T(p)=T(p)\$$. I cheated and went to the nLab for "closure operator", and there they demand that the monad be idempotent also, and that give us the equality.