Let $\mathcal D$ be a category. A familiar bit of category theory says that Grothendieck fibrations $\mathcal C \to \mathcal D$ are equivalent to pseudofunctors $F: \mathcal D^\mathrm{op} \to \mathsf{Cat}$.

Let's say that a *$\mathcal D$-fibered Grothendieck opfibration* is a functor $\mathcal B \to \mathcal C$ satsifying the following conditions:

The composite $\mathcal B \to \mathcal C \to \mathcal D$ is a Grothendieck fibration.

$\mathcal B \to \mathcal C$ preserves $\mathcal D$-cartesian morphisms.

For every $D \in \mathcal D$, the fiber functor $\mathcal B_D \to \mathcal C_D$ is a Grothendieck opfibration.

For every $f: D' \to D$ in $\mathcal D$ and $\mathcal B_D$-cocartesian morphism $g: X \to Y$ in $\mathcal C_D$, the reindexed morphism $f^\ast(g): f^\ast(X) \to f^\ast(Y)$ in $\mathcal C_{D'}$ is $\mathcal B_{D'}$-cocartesian.

**Questions:**

(0. Is there a more standard name for $\mathcal D$-fibered Grothendieck opfibrations?)

- What is the analog of the fibration - pseudofunctor correspondence in this setting?

For Question 1, the idea is that if $\mathcal C \to \mathcal D$ is a Grothendieck fibration, then a $\mathcal D$-fibered Grothendieck opfibration $\mathcal B \to \mathcal C$ should straighten to assign to every $D \in \mathcal D$ a pseudofunctor $G_D: \mathcal C_D \to \mathsf{Cat}$ which is "suitably functorial in $D$". But I'm wondering if somebody has written up the details.