This is a question about Cisinski's model structure ex nihilo, chapter 2.4 in his book Higher categories and homotopical algebra. It defines what is an exact cylinder $I$ in the presheaf category. With a class of $I$-anodyne extension this gives a model structure.
I just wonder if $IX \to X$ is a weak equivalence. It seems so for Kan-Quillen model structure and Joyal model structure for indirect reasons. I'd like to know whether this hold in general.
Initially one may hope $IX \to X$ is in fact an $I$-homotopy equivalence. But this requires to construct a natural transformation $I\circ I \to I$ that gives a homotopy between $\partial_0\circ \sigma$ and $\mathrm{id}$, which shows no clue for me.
Edit: I see that $X \to IX$ is $I$-anodyne and thus weak equivalence.
The morphism $I\otimes X\to X$ is indeed a weak equivalence. This is because $\partial I\otimes X\to I\otimes X\to X$ is a cylinder object (with Cisinski's definition of that term), so the map $\{\varepsilon\}\otimes X\to I\otimes X\to X$ is the identity on $X$ for $\varepsilon=0,1$. The map $\{\varepsilon\}\otimes X\to I\otimes X$ is indeed an $I$-anodyne map because $I$ is exact (so preserves initial objects, and then we use $\mathrm{An}1$ in Definition 2.4.11). But $\mathrm{id}_X$ is a weak equivalence as well, so by the $2$-out-of-$3$ property of weak equivalences, it follows that $I\otimes X\to X$ is also a weak equivalence.