Let's say we are given a strict $2$-category $C$. One can obtain a $1$-category from this $2$-category by deleting the $2$-cells (and forgetting horizontal composition). Similarly, one can consider the $1$-category obtained from $C$ by deleting the $0$-cells (and forgetting the horizontal composition). For instance, if $C$ is the $2$-category of (small) categories, in the latter construction we obtain the category of all functors between small categories and natural tranformations between them.
Does this latter procedure of obtaining a $1$-category from a $2$-category by deleting the $0$-cells have a name? Does it appear somewhere in the literature? It seems to me that one can generalize this procedure to strict $n$-categories, right? Namely by deleting all $k$-cells with $0\leq k \leq n-2$.