category whose hom-sets is preordered or partially orders

Question

I have a particular problem of handling category whose hom-sets are preordered (symmetric, transitive but not-always antisymmetric). In other words, I want to investigate the following situation. Let $X$ be a category and $x$, $y$ be objects of $X$. each hom-set $X(x,y)$ is preordered by $\le_{x,y}$. I am wondering if there is a general theory of such categories so that I can construct my theory on top of that.

Answer 1

Such a thing can be called a category enriched over preorders, or a locally posetal $2$-category (I guess it should be "locally preordered" but "locally posetal" is probably a better search term), or a $2$-poset (again, I guess it should be "$2$-preorder" but this is probably a better search term), or a $(1, 2)$-category, depending on taste. I'm not aware of a general theory but those might be helpful search terms.