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](https://ncatlab.org/nlab/show/axiom+of+separation#ehrhards_reformulation) 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?



Reference Request: Comprehension for multicategories
In [Comprehensive factorisation systems](https://arxiv.org/abs/1710.09438), 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](http://sqig.math.ist.utl.pt/pub/HermidaC/Fib-Yon.ps) 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.


