Let $j: K\to K′$ be a categorical equivalence of simplicial sets. By [HTT, Remark 2.1.4.11], we have a Quillen equivalence (with the covariant model structures)
$$
j\_!:\mathsf{sSet}\_{/K}\rightleftarrows\mathsf{sSet}\_{/K'}:j^\*.
$$
On the other hand, we have the functor $\mathbf{\Delta}\to\mathsf{sSet}, [n]\mapsto\Delta^n$, thus a functor $\mathbf{\Delta}\_{/K}\to\mathbf{\Delta}\_{/K'}$ on categories of simplices.


I want to know: is this functor also a categorical equivalence (i.e., equivalence of ordinary categories)?


The initial motivation for asking this is: we have $\operatorname{colim}((\mathbf{\Delta}\_{/K})^{\rm op}\xrightarrow{\*}\mathcal{S})\simeq\operatorname{colim}\_{[n]\in\mathbf{\Delta}^{\rm op}}K\_n$, viewing each $K\_n$ as a discrete/$0$-truncated object in $\mathcal{S}$ (I think, by Bousfield-Kan). I want to see if this can also be identified with $\operatorname{colim}(K\xrightarrow{\*}\mathcal{S})$ in a canonical way (in the $\infty$-category of spaces $\mathcal{S}$) so that $\operatorname{colim}\_{[n]\in\mathbf{\Delta}^{\rm op}}K\_n$ in $\mathcal{S}$ depends on $K$ only up to categorical equivalence, which I can confirm when $K$ is an $\infty$-category.


(\* means constant functor with value \*.)



Categorical equivalences vs. categories of simplices
No it is not. But there is a replacement for this. There is another model of $\infty$-categories: marked simplicial sets. If $X$ is a simplicial set with a set of marked $1$-simplices $S$, the fibrant replacement of $(X,S)$ is a model of the localization by $S$ of the $\infty$-category corresponding to $X$ (as a mere simplicial set). There is always a canonical map $\tau\_K:\Delta\_{/K}\to K$ sending an $n$-simplex over $K$ to its value at $n$. We can mark those $1$-simplices of $\Delta\_{/K}$ that are sent to the identites in $K$. Then the map $\tau\_K:\Delta\_{/K}\to K$ is an equivalence in the model structure of marked simplicial sets (where the marked simplices in $K$ are the identities). By a 2 out of 3 property, a morphism of simplicial sets $K\to K'$ is a categorical equivalence if and only if the induced functor $\Delta\_{/K}\to\Delta\_{/K'}$ is a weak equivalence of marked simplicial sets. The main properties of the comparison map $\tau\_K:\Delta\_{/K}\to K$ needed to prove what I claim above are proved in [Kerodon](https://kerodon.net/tag/01NC) or in my book [Higher categories and homological algebra](https://cisinski.app.uni-regensburg.de/CatLR.pdf) (Prop. 7.3.15) for instance. The model structure on marked simplicial sets is discussed in HTT, of course.


