nLab locally groupoidal 2-category

A $2$-category is locally groupoidal if each of its hom-categories is a groupoid. In other words, a locally groupoidal $2$-category is a $(2,1)$-category.

As the concept of $2$-category makes sense either strictly or weakly (as a bicategory), so a locally groupoidal $2$-category might be either a locally groupoidal strict $2$-category or a locally groupoidal bicategory. As for $2$-categories in general, it is really the strictness of the functors between them that matters.