Category of monoids and monoid homomorphisms

What are objects and morphisms in the category of monoids in Haskell?

The objects are types that implement Monoid type class. The morphisms are functions that satisfy some conditions.

Imagine that you have types M and N that are instances of Monoid type class. First, a morphism has to map mempty of M to mempty of N (f (mempty @M) = mempty @N). Second, it has to map values x and y of M to f x and f y of N so that f (x <> y) = f x <> f y.

Let’s take an example. We’ll pick a list of Char ([Char]) and Sum Int. Then length is a morphism from [Char] to Sum Int.

You can see both length (mempty @[Char]) and getSum (mempty @(Sum Int)) is 0. Also you can see both length (['A', 'B'] <> ['C', 'D', 'E']) and getSum (Sum (length ['A', 'B']) <> Sum (length ['C', 'D', 'E'])) is 5.

These functions are said to preserve monoidal structure and called monoid homomorphisms.