Proposition 15.3.5. Let \(C\) be an \(\infty \)-category with finite products. Then the cartesian monoidal structure \((C,\times )\) is cartesian monoidal.
Proof. The transport formula in Proposition 15.3.1 shows that the monoidal unit is the terminal object of \(C\) and the binary tensor product is the product in \(C\). Under the identification of the fiber over \(\lra {1}\) with \(C\), the two inert projections are the two maps from this product induced by the terminal maps of its factors, hence are precisely the product projections. Thus \((C,\times )\) satisfies the two conditions of Lemma 15.3.4. □
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