Lemma 16.3.3 (Pushout product). Let \(C\) be a symmetric monoidal \(\infty \)-category that admits finite colimits and such that \(X \otimes -\colon C\to C\) preserves finite colimits for every \(X\in C\). Then \(\Ar (C)\) admits a symmetric monoidal structure via Day convolution. Its unit is \(\emptyset \to \unit \), and its tensor product is the pushout product \[ (f\colon X\to Y)\mathbin {\square }(f'\colon X'\to Y') \; := \; (X\otimes Y')\sqcup _{X\otimes X'}(Y\otimes X')\longrightarrow Y\otimes Y'. \]
Proof. Equip \([1]=\{0\leq 1\}\) with the symmetric monoidal structure given by the minimum. Since the relative tensor-product slices of \(([1],\min )\) are finite, Proposition 16.2.7 applies to \(\Fun ([1],C)=\Ar (C)\). By Remark 16.2.9, the unit is the left Kan extension of \(\unit \colon *\to C\) along \(\{1\}\hookrightarrow [1]\), hence is \(\emptyset \to \unit \). The binary Day convolution is the left Kan extension along \(\min \colon [1]\times [1]\to [1]\); the pointwise formula identifies its value at \(0\) with the displayed pushout and its value at \(1\) with \(Y\otimes Y'\). □
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