Lemma 10.5.6 (Naturality in the base). The symmetric monoidal functors \(\xi _X\) assemble into a natural transformation \[ \xi \colon \Vect _{\R }^{\disc }(-)^{\simeq } \Longrightarrow \Fun (\Pi _{\infty }(-),\An _*) \] between functors \(\Top \catop \to \CMon (\Cat _{\infty })\), where the functor categories on the right carry their pointwise symmetric monoidal structures.
Proof. The assignments \(X\mapsto \Vect _{\R }^{\disc }(X)^{\simeq }\) and \(X\mapsto \mathrm {Bun}(X)\) are classified by cartesian fibrations over \(\Top \), whose cartesian morphisms are the pullback squares of vector bundles and fiber bundles, respectively. The total-space functor and fiberwise one-point compactification preserve cartesian morphisms.
Taking fundamental animae of the horizontal and vertical maps gives a functor from the cartesian fibration of fiber bundles to the codomain cartesian fibration \[ \Ar (\An )\longrightarrow \An . \] It preserves cartesian morphisms: for a continuous map \(f\colon Y\to X\) and a fiber bundle \(E\to X\), the defining pullback square for \(f^*E\) is sent by \(\Pi _{\infty }\) to a pullback square, since \(E\to X\) is a Serre fibration. After cartesian straightening, pointed realization, and pointed straightening, these cartesian functors therefore give the asserted natural transformation. The constructions preserve the fiberwise symmetric monoidal structures, so this is a natural transformation of functors valued in \(\CMon (\Cat _{\infty })\). □
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