Lemma 3.25.
Let \(F\colon T \to T'\) be a functor preserving colimits and finite limits. Then \(F\) preserves homotopy groups: for all \(X \in T\) we have a natural isomorphism
\[F(\pi_n(X)) \cong \pi_n(F(X)) \qin T_{/F(X)}.\]
Proof
Since \(F\colon T_{/X} \to T'_{/F(X)}\) preserves finite limits and \(X^{S^n}\) is a finite limit, we see that \(F(X^{S^n}) \cong F(X)^{S^n}\). Moreover, a colimit-preserving functor between presentable categories is a left adjoint. Thus \(F\) admits a right adjoint, and Lemma 3.8 shows that it commutes with \(\tau_0\).