Characterization cartesian monoidal

Lemma 15.3.4. A symmetric monoidal \(\infty \)-category \((D,\otimes )\) is cartesian monoidal if and only if the following two conditions are satisfied:

(1)

The monoidal unit \(\unit \in D\) is a terminal object;

(2)

For any two objects \(X\) and \(Y\) of \(D\), the two maps \(X \otimes Y \to X \otimes \unit \simeq X\) and \(X \otimes Y \to \unit \otimes Y \simeq Y\) induced by the maps \(X \to \unit \) and \(Y \to \unit \) from the first condition exhibit the tensor product \(X \otimes Y\) as a product of \(X\) and \(Y\).