An endofunctor on a category$\mathcal{A}$ is pointed if it is equipped with a morphism from the identity functor.

Definition

An endofunctor$S:\mathcal{A}\to \mathcal{A}$ is called pointed if it is equipped with a natural transformation$\sigma:Id_\mathcal{A} \to S$. It is called well-pointed if $S\sigma = \sigma S$ (as natural transformations $S\to S^2$).

The definition of a pointed endofunctor naturally extends to any 2-category, where we can define a pointed endomorphism as an endomorphism $s : a \to a$ equipped with a 2-cell $\sigma : 1_a \Rightarrow s$ from the identity.

A monad can be regarded as a pointed endofunctor where $\sigma$ is its unit. Such an endofunctor is well-pointed precisely when the monad is idempotent.

For pointed object in the endofunctor category

Notice that the statement which one might expect, that a pointed endofunctor is a pointed object in the endofunctor category is not quite right in general.

The terminal object of the category of endofunctors on $\mathcal{A}$ is the functor $T$ which sends all objects to $\ast$ and all morphisms to the unique morphism $\ast \to \ast$, where $\ast$ is the terminal object of the category $\mathcal{A}$. So a pointed object in the endofunctor category should be an endofunctor $S:\mathcal{A} \to \mathcal{A}$ equipped with a natural transformation $\sigma:T \to S$.

Rather, a pointed endofunctor is equipped with a map from the unit object for the monoidal structure on the endofunctor category.