This question is almost a duplicate of this one, but not quite. There the person asked about examples and intuition, I am asking about terminology and applications, and I am addressing my question more to semigroup theorists than to category theorists.
Having background in group theory, when studying category theory I find it helpful to consider restrictions of different category-theoretic definitions to the cases when the category in question is a group or a monoid (viewed as categories with a single object).
When looking into monads, I have figured out that monads in a group are exactly the inner automorphisms of the group, together with the conjugating elements. In the case of monoids, a monad in a monoid $X$ is described by an endomorphism $\theta\colon X\to X$ together with two elements $m,h\in X$ such that:
$\forall x\in X$, $\theta(x)m = m\theta(\theta(x))$,
$\forall x\in X$, $\theta(x)h = hx$,
$m^2 = m\theta(m)$,
$m\theta(h) = mh = 1$.
Do such endomorphisms of monoids have a special role in algebra? Do they have a special name?
