Proposition 2.3.9. The functor \(\Pi _{\infty }\colon \Top \to \An \) sends every pushout square in \(\Top \) whose vertices are cell complexes and one of whose maps is a relative cell complex to a pushout square in \(\An \).
Proof. This follows directly from Theorem 2.3.8 and the right exactness of localization functors stated in Corollary 2.2.9. โก
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