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A classical result due to Diers shows that a copresheaf$F\colon\mathcal{A}\to\mathbf{Set}$ on a category $\mathcal{A}$ is a coproductof representables precisely when each connected component of $F$'s category ofelements has an initial object. Most often, this condition is imposed on acopresheaf 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 apseudofunctor $F\colon\mathscr{A}\to\mathbf{Cat}$ be a coproduct ofrepresentables is often too strong of a condition. Instead, we will only askthat $F$ be a lax conical colimit of representables. This in turn allows forthe weaker notion of lax generic factorizations (and lax familialrepresentability) for pseudofunctors of bicategories$T\colon\mathscr{A}\to\mathscr{B}$.  We also compare our lax familial pseudofunctors to Weber's familial2-functors, finding our description is more general (not requiring a terminalobject in $\mathscr{A}$), though essentially equivalent when a terminal objectdoes exist. Moreover, our description of lax generics allows for an equivalencebetween 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 adjointpseudofunctors.

Lax familial representability and lax generic factorizations