Formal category theory in $\mathrm{\infty }$-equipments II: Lax functors, monoidality and fibrations

We study the framework of $\mathrm{\infty }$-equipments which is designed to produce well-behaved theories for different generalizations of $\mathrm{\infty }$-categories in a synthetic and uniform fashion. We consider notions of (lax) functors between these equipments, closed monoidal structures on these equipments, and fibrations internal to these equipments. As a main application, we will demonstrate that the foundations of internal $\mathrm{\infty }$-category theory can be readily obtained using this formalism.