Proposition 15.3.1. Let \(C\) be an \(\infty \)-category with finite products. Right-adjoint unfurling extends \(C^{(-)}\) to a finite-product-preserving functor \[ (C,\times ):=\Unf ^{\mathrm {R}}(C^{(-)})\colon \Span (\Fin )\longrightarrow \Cat _{\infty } \] which sends a span \(I\xleftarrow {f}K\xrightarrow {g}J\) to \(g_*f^*\). It defines a symmetric monoidal structure on \(C\) whose unit is the terminal object and whose tensor product is the cartesian product.
Proof. Apply Corollary 15.1.5 to \(C^{(-)}\). The resulting transport sends an \(I\)-indexed family to \[ (x_i)_{i\in I}\longmapsto \left (\prod _{k\in g^{-1}(j)}x_{f(k)}\right )_{j\in J}. \] The argument of Proposition 15.2.1 applies verbatim to show that the construction preserves finite products: the inclusion \(\Fin \catop \hookrightarrow \Span (\Fin )\) preserves finite products and is the identity on objects, so product preservation may be tested on the original functor \(C^{(-)}\). The empty product gives the terminal object as unit, and transport along \(\lra {2}\xleftarrow {=}\lra {2}\xrightarrow {\nabla }\lra {1}\) gives the binary product. The inert projections in \(C^{\times }\) induce in the fiber \(C\) the two projection maps from this product. □
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