Remark In the situation of Corollary, the Kan complex $Q_{}$ (and the anodyne morphism $f$) can be chosen to depend functorially on $X_{}$. This follows from the proof of Proposition given below, but there are other (arguably more elegant) ways to achieve the same result. For example, we can take $Q$ to be the simplicial set $\operatorname{Ex}^{\infty }(X)$ of Construction (see Propositions and, or the singular simplicial set $\operatorname{Sing}_{\bullet }(|X|)$ (see Proposition and Theorem These constructions also have non-aesthetic advantages: for example, the functors $X \mapsto \operatorname{Ex}^{\infty }(X)$ and $X \mapsto \operatorname{Sing}_{\bullet }(|X|)$ both preserve finite limits.