I like to take example categories from order theory, like with lattices and preorders and such. All such orders are also categories; the objects are the elements of the order, and the arrows are the relationships. However, if x <= y in the order, then there's only one arrow x -> y -- the order doesn't distinguish the different ways two objects can be related, only the mere fact that they are. (This is the difference between an order and a category: categories do allow us to distinguish the multiple ways two things can be related.)
Lots of category theory has direct analogues in order theory. Functors are monotone functions, for instance; monads are closure operators; and presheafs are lower sets. But since there's at most one arrow between any two objects, you don't have to worry about all the coherence conditions you end up seeing in category theory. Their whole purpose is to make sure that you pick "the right" arrow in various circumstances; but when there's only one arrow, it is trivially the right one.
Lots of category theory has direct analogues in order theory. Functors are monotone functions, for instance; monads are closure operators; and presheafs are lower sets. But since there's at most one arrow between any two objects, you don't have to worry about all the coherence conditions you end up seeing in category theory. Their whole purpose is to make sure that you pick "the right" arrow in various circumstances; but when there's only one arrow, it is trivially the right one.