Formal category theory in $\mathrm{\infty }$-equipments I

We generalize proarrow equipments from strict category theory to the $\mathrm{\infty }$-categorical setting, introducing the concept of $\mathrm{\infty }$-equipments. These are specific double $\mathrm{\infty }$-categories that support an internal higher category theory. This paper explores several examples of $\mathrm{\infty }$-equipments, including the prototypical example of the $\mathrm{\infty }$-equipment of $\mathrm{\infty }$-categories and the more general $\mathrm{\infty }$-equipments of internal $\mathrm{\infty }$-categories. The ultimate objective of this article is to study the basic concepts of category theory within an arbitrary $\mathrm{\infty }$-equipment, such as colimits and Kan extensions.