**Background:**

A pointed object $X$ in a category $C$ with terminal object $*$ is a map $*\rightarrow X$. Such objects with basepoint-preserving maps form their own category of pointed objects $C^{*/}$. There is a canonical forgetful functor $U:C^{*/}\rightarrow C$ that forgets the basepoint. Furthermore, this has a left adjoint $(-)_{+}:C\rightarrow C^{*/}$ which sends an object $Y$ to the coproduct $Y\coprod *$ equipped with the canonical basepoint inclusion. The adjunction $(-)_{+}\dashv U$ induces a monad on $C$ (think this is called the ``maybe monad"). The category of algebras over this monad is $C^{*/}$.

**Question:**

There is also a notion of a "multi pointed object:" an object $X$ equipped with a map from a coproduct of the terminal object with itself a bunch of times. The objects with the obvious maps form a category $C_{multi}$ . Does this category arise as a category of algebras over some sort of "maybe" monad?

**Edit**

To clarify: the objects are objects in $C_{multi}$ with a fixed (we can even assume finite) number of basepoints. The morphisms are maps that preserve those basepoints.