are there examples of "category-like" structures where distinct pairs of objects have hom-sets that aren't disjoint?

Question

I understand (based on the relatively few examples of categories I have at my disposal), why distinct pairs of objects should have disjoint hom-sets, but I wanted to know of any structures that are almost categories in the sense that they satisfy all the axioms for a category except that one.

Answer 1

Well, sets and maps. In set theory, a map from $A$ to $B$ is a certain subset of $A \times B$. If the map factors through some subset $B'$ of $B$, we get two maps $A \to B$ and $A \to B'$ which are equal as sets.

For example, for a set theorist, $\emptyset$ is a map from $\emptyset$ to any set. But for a category theorist, a morphism has to have a specified domain and codomain.

For more on this "disjoint-axiom", see SE/651464