What is the "Unit Element" in a Categorical Monad?

Question

From the wikipedia definition of a categorical monoid:

Question: What is the nature of $I$? Say, in the context of Set?

Answer 1

To understand what a monoid is, you should first understand what a monoidal category is. Roughly, it's a category $C$ together with a bifunctor $\otimes : C \times C \to C$ satisfying certain associativity properties. One also requires $C$ to have a special object $I$ that acts as an identity for $\otimes$. A monoid $M$ in $C$ is one that's "closed" under $\otimes$. This is made precise by saying that $M$ should satisfy the axioms above.

Now $Set$ is made into a monoidal category by taking the functor $\otimes$ to be the Cartesian product. The identity should be any one point set (all of them are uniquely isomorphic). A monoid in $Set$ is a set $A$ together with an associative binary operation $A \times A \to A$ and an element $e \in A$ that acts as a neutral element for the binary operation (we may think of $e$ as the inclusion $\{e\} \to A$). In other words, a monoid in $Set$ is what you usually understand to be a monoid. For example, the natural numbers together with addition form a monoid.