- 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:
How these two definitions corresponds to each other.
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!