Warning 3.6.3.9. In the formulation of Proposition 3.6.3.8, the hypothesis that $X$ and $Y$ have the homotopy type of a CW complex cannot be omitted. For any topological space $Y$, the counit map $v: | \operatorname{Sing}_{\bullet }(Y) | \rightarrow Y$ is a weak homotopy equivalence (Corollary 3.6.4.2), whose domain is a CW complex (Remark 1.2.3.12). If $Y$ satisfies the conclusion of Proposition 3.6.3.8, then $v$ is a homotopy equivalence, so $Y$ has the homotopy type of a CW complex.
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