Bridgeable Double Category

A Double Category $\mathcal{D}\rightarrowtriangle C\times C$ is bridgeable when

• The right total category $\int_{\mathcal{C}}\mathcal{D}\rightarrowtriangle\mathcal{C}$ is a Cartesian Fibration
• The functor $i:\mathcal{C}\to\int_{\mathcal{C}}\mathcal{D}$ that picks out loose identities is a Fibred Functor.

If we unfold this definition, we can see that this lets us pull back loose cells in a suitably universal way.

Questions

• Is the Double Category of Mates bridgeable when we restrict the 1-cells to be Discrete Cartesian Fibrations?

References

• Nathanael Arkor: “A (Virtual) Double Category Theorist’s Perspective on Polynomials,” 2025. https://www.youtube.com/watch?v=tCbRfjv6JQ4.