The other day I noticed the following. Suppose you have a space X; say a metric space. Let C(X) be the completion of X. We have the obvious injection X ⇒ C(X) which is a continuous function. Any continuous function can be naturally lifted to a continuous function on the completed spaces so we have a function (X ⇒ Y) ⇒ (C(X) ⇒ C(Y)). Finally the completion of a completed space is the same so there is a continuous function C(C(X)) ⇒ C(X).
Everywhere I look I see monads. Or as pfloide would say, “Everywhere you look you see adjoint functors”.