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}. \$