Proposition 15.2.1. Let \(C\) be an \(\infty \)-category with finite coproducts. The unfurling of \(C^{(-)}\) is a finite-product-preserving functor \[ (C,\amalg ):=\Unf ^{\mathrm {L}}(C^{(-)})\colon \Span (\Fin )\longrightarrow \Cat _{\infty } \] which defines a symmetric monoidal structure on \(C\) whose unit is the initial object and whose tensor product is the coproduct.
Proof. By Corollary 15.1.4, the functor \(C^{(-)}\) extends to \(\Span (\Fin )\). To check whether it preserves finite products, recall from part (1) of Lemma 13.3.8 that the inclusion \(\Fin \catop \hookrightarrow \Span (\Fin )\) preserves finite products. Since this inclusion is the identity on objects, we may test whether the unfurling preserves finite products after restricting along it. By the last statement of Corollary 15.1.4, the resulting functor is the original functor \(C^{(-)}\), which preserves finite products by construction. This finishes the construction of the monoidal structure.
Given the description of coproducts in terms of left adjoints, we see that transport along \(\emptyset \to \lra {1}\) selects the initial object, while transport along the active span \(\lra {2}\xleftarrow {=}\lra {2}\xrightarrow {\nabla }\lra {1}\) is the binary coproduct. □
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