Homotopy groups preserve filtered colimits

Proof. By Lemma 4.3.6, \(f\) is an isomorphism if and only if each map \(f_n\colon X_n \to Y_n\) of animae is an isomorphism. By the Whitehead Theorem for animae, Proposition 2.4.22, this is the case if \(f_n\) induces isomorphisms on all homotopy groups. By Exercise 4.4.23 this condition is equivalent to the condition that \(f\) induces isomorphisms on all homotopy groups. □