Categorical notions of fibration

Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^{op}⇝\mathbf{C}\mathbf{a}\mathbf{t}$, while the special case of discrete fibrations encode presheaves $B^{op} \to {\bf Set}$\mathrm{.}Atwo-sideddiscretevariationencodesfunctors$B^{op} \times A \to {\bf Set}$,whicharealsoknownasprofunctorsfrom$A$to$B$. By work of Street, all of these fibration notions can be defined internally to an arbitrary 2-category or bicategory. While the two-sided discrete fibrations model profunctors internally to $\mathbf{C}\mathbf{a}\mathbf{t}$, unexpectedly, the dual two-sided codiscrete cofibrations are necessary to model $\mathcal{V}$-profunctors internally to $\mathcal{V}$-$\mathbf{C}\mathbf{a}\mathbf{t}$.