Bi-initial objects and bi-representations are not so different

We introduce a functor $\mathcal V\colon \mathrm{DblCat}_{h,nps}\to \mathrm{2Cat}_{h,nps}$ extracting from a double category a $2$-category whose objects and morphisms are the vertical morphisms and squares. We give a characterisation of bi-representations of a normal pseudo-functor $F\colon \mathbf C^{\operatorname{op}}\to \mathrm{Cat}$ in terms of double bi-initial objects in the double category $\mathbb{E}l(F)$ of elements of $F$, or equivalently as bi-initial objects of a special form in the $2$-category $\mathcal V\mathbb{E}l(F)$ of morphisms of $F$. Although not true in general, in the special case where the $2$-category $\mathbf C$ has tensors by the category $\mathbf{2}=\{0\to 1\}$ and $F$ preserves those tensors, we show that a bi-representation of $F$ is then precisely a bi-initial object in the $2$-category $\mathbf{E}l(F)$ of elements of $F$. We give applications of this theory to bi-adjunctions and weighted bi-limits.