Let
$\mathrm{Top}$: the category where objects are topological spaces and morphisms are continuous maps.
$\mathrm{hTop}$: the category where objects are topological spaces and morphisms are homotopy class of continuous maps.
$\mathrm{Ho}:\mathrm{Top}\rightarrow \mathrm{hTop}$: the functor sending $f$ to $[f]$ for any continuous map $f$.
I'm beginner of homotopy theory and learning the notion of homotopy colimit from the CH8 of "Cubical Homotopy Theory" by Munson-Volić. In that book, the authors develop the theory of homotopy colimit for the category $\mathrm{Top}$, but I am looking for good textbook/reference for the homotopy colimit for the category $\mathrm{hTop}$.
For example, I want to find a reference having analogous version of the following Theorem 8.3.7 in Munson-Volić's book:
"Let $F,G:\mathrm{I}\rightarrow\mathrm{Top}$ be two functors and $N:F\rightarrow G$ are natural transformation such that $F(i)\rightarrow G(i)$ is homotopy equivalence for any $i\in \mathrm{I}$. then, the natural map $\mathrm{hocolim}F\rightarrow\mathrm{hocolim}G$ is homotopy eauivalence."
I'm expecting the similar result will still hold even if $N$ is a natural transformation in $\mathrm{hTop}^{\mathrm{I}}$, from $\mathrm{Ho}\circ F$ to $\mathrm{Ho}\circ G$. But I couldn't find any good references. I did some googleing, but the references I found were too advanced or abstract to me. So, I hope to know what are some good (accessible to beginners, requiring less background material, having many examples,...) resources for the homotopy colimit of $\mathrm{hTop}$ category.