Sorry, I am new to category theory (actually all fields of math...). When I was learning the concept of monoid in college, the identity element is roughly defined as "an element e of set which satisfy $e * a = a = a * e$, for every a in that set". But when I am learning this concept again for understanding monad. The identity element of monoid is defined as a morphism (see here). Could anyone please explain why these two definitions of monoid (in particular the identity of monoid) are equivalent. Thanks!
EDIT:
Some other definition that represents the identity of monoid as morphism:
A definition from a math website
Another definition in stackoverflow (in the highest vote answer)
The definition you linked to is doing more than just defining a monoid. It's actually defining a monoid object in a (monoidal) category. The usual notion of a monoid is a monoid object in $\operatorname{Set}$. Let's see how the definition plays out in that context.
According the the linked definition, the identity is a morphism from $1$—the unit object of the monoidal category—into the object of the monoid, $M$. In $\operatorname{Set}$, the unit object is any one-element set. The only data associated with a function from a one-element set to $M$ is knowledge of where the function sends that one element. That is, a function from a one-element set to $M$ is essentially just an element of $M$, so the definition aligns with the usual one.
The reason the general definition is phrased in terms of morphisms is because, in a general monoidal category, there might not be a notion of an "element." A morphism from the unit object to $M$ will still make sense, however.