  The paper focuses on investigating how certain relations between strict
$n$-categories are preserved in a particular implementation of
$(\infty,n)$-categories, given by saturated $n$-complicial sets. In this model,
we show that the $(\infty,n)$-categorical nerve of $n$-categories is
homotopically compatible with $1$-categorical suspension and wedge. As an
application, we show that certain pushouts encoding composition in
$n$-categories are homotopy pushouts of saturated $n$-complicial sets.
