I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory. From the general stuff about Kan-extensions (HTT 4.3.2.6) follows that taking the colimit gives a functor $\operatorname{Map}(\mathcal{C}, \mathcal{D}) \to \operatorname{Map}(\mathcal{C}^\rhd, \mathcal{D})$. But it seems to me that what one actually wants is something stronger that also takes into account maps between the diagram categories. More precisely my questions are as follows:

Where this came up is the following situation: I have two functors $f, f' \colon \mathcal I \to \mathcal J$ between (small) ordinary categories, a natural transformation $\eta \colon f \Rightarrow f'$, and a diagram $D \colon \mathcal J \to \mathcal C$ in a cocomplete $\infty$-category $\mathcal C$. Is then $f'_* \circ (D \circ \eta)_*$ homotopic to $f_*$ as maps $\operatorname{colim}_{\mathcal I} (D \circ f) \to \operatorname{colim}_{\mathcal J} D$?

The above seems like it should be a small part of a much more general statement. Namely that there is a functor $(\mathbf{Cat}_\infty)_{/\mathcal{C}} \to \mathcal{C}$ given by taking the colimit (or something more sophisticated also taking into account the cones), where we consider $\mathbf{Cat}_\infty$ as an $(\infty, 2)$-category such that $(\mathbf{Cat}_\infty)_{/\mathcal{C}}$ becomes an $(\infty, 1)$-category (here we probably need to be a little careful in which direction the 2-cells are pointing, which should depend on whether we are looking at colimits or limits).

Do these statements make sense and if so, are they true? Furthermore, are they implied by statements in HTT? I tried to look through the book, but the only thing I found that relates to these questions is 4.2.2.7 and I don't see how to extract statements as above from this.

**EDIT**: As noted by Dylan Wilson in the comments 2. is part of a paper by Mazel-Gee. However, I don't see how that version of the statement implies 1. since it seems to me that the diagram corresponding to 1. does not commute in Mazel-Gee's $\mathcal Lax(\mathcal C)$ which is the domain of the global colimit functor for $\mathcal C$ constructed there.