I recently came across the notion of comprehension for a fibration of categories $p:E\to B$ that subsumes the axiom of separation and the grothendieck construction for fibred categories. A nice definition can be found in the n-lab article on the axiom of separation under the section "Ehrhard’s reformulation".
I guess the definition can be lifted to multicategories. Does anyone know of an article where this was done?
In Comprehensive factorisation systems, Berger and Kaufmann establish a correspondence between certain orthogonal factorisation systems (the motivating example being the discrete fibration/final functor OFS on $\mathbf{Cat}$) and consistent comprehension schemes. In Section 2, they explicitly cover the setting of small multicategories. Hermida's Fibrations and Yoneda structure for multicategories contains similar ideas.
Note that these references describe only the setting of discrete fibrations. I'm not aware of a reference that covers the more general setting of (arbitrary) fibrations and comprehension for multicategories.
