Lax familial representability and lax generic factorizations

Abstract: A classical result due to Diers shows that a copresheaf $F\colon\mathcal{A}\to\mathbf{Set}$ on a category $\mathcal{A}$ is a coproduct of representables precisely when each connected component of $F$'s category of elements has an initial object. Most often, this condition is imposed on a copresheaf of the form $\mathcal{B}\left(X,T-\right)$ for a functor $T\colon\mathcal{A}\to\mathcal{B}$, in which case this property says that $T$ admits generic factorizations at $X$, or equivalently that $T$ is familial at $X$. Here we generalize these results to the two-dimensional setting, replacing $\mathcal{A}$ with an arbitrary bicategory $\mathscr{A}$, and $\mathbf{Set}$ with $\mathbf{Cat}$. In this two-dimensional setting, simply asking that a pseudofunctor $F\colon\mathscr{A}\to\mathbf{Cat}$ be a coproduct of representables is often too strong of a condition. Instead, we will only ask that $F$ be a lax conical colimit of representables. This in turn allows for the weaker notion of lax generic factorizations (and lax familial representability) for pseudofunctors of bicategories $T\colon\mathscr{A}\to\mathscr{B}$. We also compare our lax familial pseudofunctors to Weber's familial 2-functors, finding our description is more general (not requiring a terminal object in $\mathscr{A}$), though essentially equivalent when a terminal object does exist. Moreover, our description of lax generics allows for an equivalence between lax generic factorizations and lax familial representability. Finally, we characterize our lax familial pseudofunctors as right lax $\mathsf{F}$-adjoints followed by locally discrete fibrations of bicategories, which in turn yields a simple definition of parametric right adjoint pseudofunctors.