Lemma 3.26.
The functor \(\pi_n \colon T \to T\) preserves finite products.
Proof
Cotensoring by \(S^n\) preserves products, and Lemma 3.9 shows that \(0\)-truncation preserves products in every slice topos. Applying the definition therefore gives
\[\pi_n(X\times Y) \cong \pi_n(X)\times\pi_n(Y).\]