Is there a name for or a body of work about categories for which there are exactly two morphisms $f,g$ between any two objects $X,Y$? This is disrgarding self maps for which there is only the identity.
$$f: X \rightarrow Y$$ $$g: Y \rightarrow X$$
Another way to put it is "there is only one arrow in any one direction", where a direction is given $[A,B]$ a list which has a first and then a second element and the direction is first to second.
A category $C$ is called indiscrete/chaotic/codiscrete when, for each pair of objects $x, y \in C$, the hom-class $C(x, y)$ has exactly one element.
Note that, as a structure in its own right, an indiscrete category is equivalent to a set (the morphisms carry no information, because they are uniquely determined). However, indiscrete categories can be useful tools in studying other concepts. For instance, a category enriched in a bicategory is a lax functor from an indiscrete category.