Tensor products from standard presentations

Remark 4.4.20. Let \(X\) and \(Y\) be spectra. The standard presentations of \(X\) and \(Y\) provide the following description of their tensor product: \[ X \otimes Y \quad \cong \quad \colim _n \colim _m (\Sigma ^{\infty -n} X_n \otimes \Sigma ^{\infty - m} Y_m) \quad \simeq \quad \colim _n \colim _m \Sigma ^{\infty - n - m} (X_n \wedge Y_m), \] where the last equivalence holds by the previous lemma. However, this formula should be used with care: it makes implicit use of the additivity of the shift functors \(X[-n][-m] \cong X[-(n+m)]\) from Warning 4.2.11. To avoid introducing sign errors, we will not work with this formula for the tensor product.