On lax transformations, adjunctions, and monads in $(\mathrm{\infty },2)$-categories

We use the basic expected properties of the Gray tensor product of $(\mathrm{\infty },2)$-categories to study (co)lax natural transformations. Using results of Riehl-Verity and Zaganidis we identify lax transformations between adjunctions and monads with commutative squares of (monadic) right adjoints. We also identify the colax transformations whose components are equivalences (generalizing the “icons” of Lack) with the 2-morphisms that arise from viewing $(\mathrm{\infty },2)$-categories as simplicial $\mathrm{\infty }$-categories. Using this characterization we identify the $\mathrm{\infty }$-category of monads on a fixed object and colax morphisms between them with the $\mathrm{\infty }$-category of associative algebras in endomorphisms.