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 $y$.

The objects in the hom-category $C(x,y)$ are the 1-morphisms in $C$ from $x$ to $y$, while the morphisms in the hom-category $C(x,y)$ are the 2-morphisms of $C$ that are horizontally between $x$ 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.