This is both a continuation of my previous talk and an independent exposition on the subject of studying lax limits (also known as sections) of the families of categories (understood as fibrations in a flexible sense) with homotopical structure, such as model categories. The first large class of examples of such families are Quillen presheaves: those are diagrams of model categories and Quillen adjunctions. These families often appear in geometry. For Quillen presheaves one can prove that taking lax limits commutes with forming the infinity (Dwyer-Kan) localisation; we may explain what a proof of this statement looks like.
Another example of model categorical families appears in the context of algebra, and is not given by Quillen presheaves. Studying lax limits of such families can be still done model-categorically by introducing the notion of a Segal section. As a consequence one can study model-categorically various objects appearing in higher algebra, e.g. E_2 algebras over a field of positive characteristic, with statements such as the existence of homotopy colimits of these objects proven rather easily (I will be curious to know why).