What is monoid homomorphism exactly?

Question

I\'ve read about monoid homomorphism from Monoid Morphisms, Products, and Coproducts and could not understand 100%.

The author says (emphasis original):

Answer 1

Does he mean, the datatype String and Int are monoid.

No, neither String nor Int are monoids. A monoid is a 3-tuple (S, ⊕, e) where ⊕ is a binary operator ⊕ : S×S → S, such that for all elements a, b, c∈S it holds that (a⊕b)⊕c=a⊕(b⊕c), and e∈S is an "identity element" such that a⊕e=e⊕a=a. String and Int are types, so basically sets of values, but not 3-tuples.

The article says:

Let's take the String concatenation and Int addition as example monoids that have a relationship.

So the author clearly also mentions the binary operators ((++) in case of String, and (+) in case of Int). The identities (empty string in case of String and 0 in case of Int) are left implicit; leaving the identities as an exercise for the reader is common in informal English discourse.

Now given that we have two monoid structures (M, ⊕, e_m) and (N, ⊗, e_n), a function f : M → N (like length) is then called a monoid homomorphism [wiki] given it holds that f(m₁⊕m₂)=f(m₁)⊗f(m₂) for all elements m₁, m₂∈M and that mapping also preserves the identity element: f(e_m)=e_n.

For example length :: String -> Int is a monoid homomorphism, since we can consider the monoids (String, (++), "") and (Int, (+), 0). It holds that:

1. length (s1 ++ s2) == length s1 + length s2 (for all Strings s1 and s2); and
2. length "" == 0.