Trying to understand monoids in Set as general monoid in monoidal category

Question

• In the definition of a monoid in monoidal category, unit is defined as η: I → M, such that the following graph commutes.

• While in the definition of a monoid in Set, unit of a set $S$ is defined as an element $e \in S$, such that $\forall a \in S, e \cdot a = a = a \cdot e$

I would like to know:

1. How these two definitions corresponds to each other.

2. I know that $\otimes$ means the bifunctor in the definition of a monoidal category, thus $I \otimes M$, $M \otimes M$, etc. are all objects in that monoidal category. But what does $\eta \otimes 1$ on the arrow mean? Since $\eta$ is a morphism instead of an object in the monoidal category, I cannot come up with an explanation of its meaning.

Thanks a lot in advance!

Answer 1

For 1: In $\mathsf{Set}$, the monoidal product $\otimes$ is actually the Cartesian product $\times$ of sets, and $I$, the unit object, is some one-element set $1 = \{*\}$. Then if $M$ is a monoid, $\mu : M \times M \to M$ is the monoid multiplication, $\eta: 1 \to M$ is the function that sends $*$ to the unit element $e \in M$. The functions $\lambda,\rho$ give the obvious isomorphisms $M \times 1 \cong M \cong 1 \times M$. The first triangle says that $\lambda(*,m) = \mu(\eta(*),\mathrm{id}_{M}(m))$. Just filling in definitions, this gives $m = e \cdot m$. The other triangle similarly gives $m = m \cdot e$.

For 2: A bi-endofunctor on $\mathsf C$ is really just a functor $\otimes: \mathsf C \times \mathsf C \to \mathsf C$. In particular, it should take morphisms of $\mathsf C \times \mathsf C$ to morphisms of $\mathsf C$. Morphisms of $\mathsf C \times \mathsf C$ from $(X,X')$ to $(Y,Y')$ are pairs of morphisms $(f,f')$ where $f: X \to Y$ and $f': X'\to Y'$. Typically, we write $\otimes(f,f')$ as $f \otimes f'$. Thus, $\eta \otimes 1$ is the image of $(\eta,1)$ under the functor $\otimes$. Note that $1$ here refers to $\mathrm{id}_M$.