Remark 1.5.5.2. Suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix@R =40pt@C=40pt{ X' \ar [d]^{q'} \ar [r] & X \ar [d]^{q} \\ Y' \ar [r] & Y. } \]
If $q$ is a trivial Kan fibration, then so is $q'$ (this follows from Proposition 1.5.4.5, applied to the opposite of the category $\operatorname{Set_{\Delta }}$).