Free bifibrations of $(\mathrm{\infty },2)$-categories, 2-simplicial objects and the walking adjunction

In this work, we develop a fibrational approach to freely adjoining adjoints in an $(\mathrm{\infty },2)$-category. We construct the universal bifibration obtained from a cocartesian fibration of $(\mathrm{\infty },2)$-categories by adjoining cartesian lifts over a chosen class of $1$-morphisms in the base. We then use this construction to provide an explicit model for freely adjoining adjoints, together with a zig-zag formula for the resulting mapping $(\mathrm{\infty },1)$-categories. As applications, we give a model-independent proof of the universal property of the walking adjunction, providing an alternative proof of a theorem of Riehl–Verity, and establish the universal characterization of the simplex $2$-category conjectured by Dyckerhoff–Kapranov–Schechtman–Soibelman.