Lemma 15.2.22. A symmetric monoidal \(\infty \)-category \((D,\otimes )\) is cocartesian monoidal if and only if the following two conditions are satisfied:
- (1)
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The monoidal unit \(\unit \in D\) is an initial object;
- (2)
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For any two objects \(X\) and \(Y\) of \(D\), the two maps \(X \simeq X \otimes \unit \to X \otimes Y\) and \(Y \simeq \unit \otimes Y \to X \otimes Y\) induced by the maps \(\unit \to X\) and \(\unit \to Y\) from (1) exhibit the tensor product \(X \otimes Y\) as a coproduct of \(X\) and \(Y\).
Proof. In light of the equivalence \[ \Mm _D(\{x_i\}_{i \in I};y) \simeq \Hom _D(\bigotimes _{i \in I} x_i, y), \] this is an immediate consequence of Corollary 15.2.18, together with Proposition 15.2.8. □
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