I would call such a monoid an *idempotent monoid* since \\(f\\) follows the rule, \\(f\circ f = f \\).

Edit: Or more generally, if we let,
\\[\large
f^{\circ n} = \underbrace{f\circ f\circ f\circ \cdots f}_{n \text{ times}}
\\]

then an *idempotent monoid* is a monoid with a unique morphism,
\\[\large
f = f^{\circ n}, \text{for some } n \in \mathbb{N}.
\\]