Limits of (∞, 1)-categories with structure and their lax morphisms

Riehl and Verity have established that for a quasi-category A that admits limits, and a homotopy coherent monad on A which does not preserve limits, the Eilenberg-Moore object still admits limits; this can be interpreted as a completeness result involving lax morphisms. We generalise their result to different models for (∞, 1)-categories, with an abundant variety of structures. For instance, (∞, 1)-categories with limits, Cartesian fibrations between (∞, 1)-categories, and adjunctions between (∞, 1)-categories. In addition, we show that these (∞, 1)-categories with structure in fact possess an important class of limits of lax morphisms, including ∞-categorical versions of inserters and equifiers, when only one morphism in the diagram is required to be structure-preserving. Our approach provides a minimal requirement and a transparent explanation for several kinds of limits of (∞, 1)-categories and their lax morphisms to exist.