Consider a functor $F: I \to (\mathbf{Cat}_\infty)_{/K}$, then the unique map $F \Rightarrow \mathrm{Const}_K$ induces a functor $\int F \to I \times K$.
1) Is $\int F \to I \times K$ a bifibration?
2) Does the construction in Notation 4.2.3.1. in HTT give a model for the corresponding cylinder, i.e. it’s collage?
3) Is this explained anywhere in more detail?