[categories] Question: double category of fibrations

Dear all, I am wondering about a double category of fibrations. In particular, whether the usual 2-category Fib (fibrations, cartesian functors, and natural transformations) extends to a double category, and if so what the notion of promorphisms and cells would be. I'm thinking that the promorphisms might be something like 2-sided fibrations (as described by e.g. Loregian and Riehl). Has anyone worked this out? I am very much looking forward to your responses! All the best, Sarah