Products of localizations

Exercise 1.5.24. Let \(C\) and \(D\) be \(\infty \)-categories with collections of morphisms \(W\) and \(W'\), respectively, and assume that both collections contain all identity morphisms. Use the universal property to show that \[ (C \times D)[(W \times W')^{-1}] \iso C[W^{-1}] \times D[{W'}^{-1}]. \] Deduce that \(\geom {C \times D} \iso \geom {C} \times \geom {D}\).