Proposition 1.5.5.11. Let $p: X \rightarrow Y$ be a trivial Kan fibration of simplicial sets. Then:
If $X$ is a Kan complex, then $Y$ is a Kan complex.
If $X$ is a contractible Kan complex, then $Y$ is a contractible Kan complex.
If $X$ is an $\infty $-category, then $Y$ is an $\infty $-category.
Proof. We will prove $(1)$; the proofs of $(2)$ and $(3)$ are similar. Suppose we are given a pair of integers $0 \leq i \leq n$ with $n > 0$; we wish to show that every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow Y$ can be extended to an $n$-simplex of $Y$. Since $p$ is a trivial Kan fibration, we can write $\sigma _0 = p \circ \tau _0$ for some morphism $\tau _0: \Lambda ^{n}_{i} \rightarrow X$ (Proposition 1.5.5.4). If $X$ is a Kan complex, we can extend $\tau _0$ to an $n$-simplex $\tau $ of $X$. Then $\sigma = p \circ \tau $ is an $n$-simplex of $Y$ satisfying $\sigma _0 = \sigma |_{ \Lambda ^{n}_{i} }$. $\square$