  We introduce a generalization of proarrow equipments to the
$\infty$-categorical setting, termed $\infty$-equipments. These equipments take
the form of particular double $\infty$-categories that support a notion of
internal formal higher category theory. We will explore several examples of
$\infty$-equipments. In particular, we will study the prototypical example
given by the $\infty$-equipment of $\infty$-categories, and construct
$\infty$-equipments of $\infty$-categories internal to general $\infty$-toposes
as well. The final objective of this article is to study the fundamental
notions of formal higher category theory within an arbitrary
$\infty$-equipment. To this end, we not only follow the classical approach but
also expand upon it.
