simplicial stuff - Why is the induced map between pullbacks (of inclusions) by a right fibration a deformation retract?

Let $X$ be a simplicial set. Let $Xto Delta^n$ 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 $i: 0leq ileq n$) be the obvious inclusion maps.

Then why is the induced map:

$$Xtimes_{Delta^n} Delta^{{n-i}} hookrightarrow Xtimes_{Delta^n}Delta^{{n-i,dots, n}}$$

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.)