Comparing lax functors of $(\mathrm{\infty },2)$-categories

In this work, we study oplax normalised functors of $(\mathrm{\infty },2)$-categories. Our main theorem is a comparison between the notion of oplax normalised functor of scaled simplicial sets due to Gagna-Harpaz-Lanari and the corresponding notion in the setting of complete Segal objects in $(\mathrm{\infty },1)$-categories studied by Gaitsgory and Rozenblyum. As a corollary, we derive that the Gray tensor product of $(\mathrm{\infty },2)$-categories as defined by Gaitsgory-Rozenblyum is equivalent to that of Gagna-Harpaz-Lanari. Moreover, we construct an $(\mathrm{\infty },2)$-categorical variant of the quintet functor of Ehresmann, from the $(\mathrm{\infty },2)$-category of $(\mathrm{\infty },2)$-categories to the $(\mathrm{\infty },2)$-category of double $(\mathrm{\infty },1)$-categories and show that it is fully faithful. As a key technical ingredient, given $(\mathbb{C},E)$ an $(\mathrm{\infty },2)$-category equipped with a collection of morphisms and a functor of $(\mathrm{\infty },2)$-categories $f:\mathbb{C}\to \mathbb{D}$, we construct a right adjoint to the restriction functor $f^{\ast }$ from the $(\mathrm{\infty },2)$-category of functors $\mathbb{D}\to \mathbb{C}\text{\hspace{0.17em}⁣}{\mathrm{at}}_{(\mathrm{\infty },2)}$ and natural transformations to the $(\mathrm{\infty },2)$-category of functors $\mathbb{C}\to \mathbb{C}\text{\hspace{0.17em}⁣}{\mathrm{at}}_{(\mathrm{\infty },2)}$ and partially lax (according to $E$) natural transformations. We apply this new technology of partially lax Kan extensions to the study of complete Segal objects in $(\mathrm{\infty },1)$-categories and double $(\mathrm{\infty },1)$-categories which allows us to define the notion of an enhanced Segal object (resp. enhanced double $(\mathrm{\infty },1)$-category), the former yielding yet another model for the theory of $(\mathrm{\infty },2)$-categories.