Definition of the smash-product monoidal structure

Definition 16.3.5. Let \(C\) be an \(\infty \)-category with finite products and finite colimits such that the functor \(X \times -\colon C \to C\) preserves finite colimits for all \(X \in C\). We may then apply Lemma 16.3.4 to the cartesian monoidal structure on \(C\). We refer to the resulting symmetric monoidal structure \((C_*,\wedge ,S^0)\) on the pointed objects in \(C\) as the smash product. The monoidal unit is \(S^0 := * \sqcup *\), while the smash product \(X \wedge Y\) is given by \[ X \wedge Y \quad := \quad \cofib (X \vee Y \to X \times Y), \] where \(X \vee Y := X \sqcup _* Y\).