[catnaroek]

Some examples of categories whose arrows aren't functions:

(0) Any monoid can be viewed as a category with a single object. The arrows from the object to itself are the monoid's elements.

(1) Any preorder can be viewed as a category such that, between any two objects, there is at most one arrow, precisely when the source is less or equal than the target.

(2) Given a directed graph, or more generally a quiver[0], there is a small category[1] whose objects are the quiver's nodes, and whose arrows are the paths (finite sequences of edges) from a source node to a target node.

[0] https://en.wikipedia.org/wiki/Quiver_(mathematics)

[1] https://en.wikipedia.org/wiki/Free_category