A -category is locally posetal or locally partially ordered or Pos-enriched if every hom-category is a poset - an object of the category Pos of partial orders. One can also consider a locally preordered -category, where every hom-category is a proset (a preordered set); up to equivalence of -categories, these aren't any more general.
Locally posetal -categories are the usual model of 2-posets, aka (1,2)-categories. Just as the motivating example of a -category is the -category Cat of categories, so the motivating example of a -poset is the -poset Pos of posets. If you interpret as a full sub-2-category of , then it is indeed locally posetal. Similarly, the -category of prosets is a locally preordered -category that is equivalent to .
Compare the notion of partially ordered category. A locally partially ordered category is a category enriched over the category Pos of posets, while a partially ordered category is a category internal to . Similarly, a locally partially ordered category is a special kind of -category, while a partially ordered category is a special kind of double category.