Let $X$ be a simplicial set. Let $X\to \Delta^n$ be a right fibration (has the right lifting property with respect to right horn inclusions), and let $$\Delta^{\{n-i\}}\hookrightarrow \Delta^{\{n-i,\dots, n\}}\hookrightarrow \Delta^n$$ (for a fixed $i: 0\leq i\leq n$) be the obvious inclusion maps.

Then why is the induced map:

$$X\times_{\Delta^n} \Delta^{\{n-i\}} \hookrightarrow X\times_{\Delta^n}\Delta^{\{n-i,\dots, n\}}$$

a deformation retract? It's not like we can apply Whitehead's theorem, since $\Delta^n$ is not a Kan complex in general.

(This is from the end of the proof of proposition 2.2.3.1 of HTT. The statement should be true out of the context in the book with the hypotheses I've given, but if not, there's the source.)