Page "Functor" Paragraph 49

A small category with a single object is the same thing as a monoid: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation.

Functors between one-object categories correspond to monoid homomorphisms.

So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.