nLab
hom-category

Recall that in a category $C$, each pair of objects $x$ and $y$ determines a set $C(x,y)$, the hom-set of $x$ and $y$. In a $2$-category $B$, each pair of objects determines a category $B(x,y)$. This category is the hom-category of $x$ and $y$.

As a $2$-category is enriched over Cat, a hom-category is a special case of a hom-object. But the hom-category makes sense also for the weakly enriched concept of bicategory.