Lemma 10.5.2 (Pointwise smash products). For every anima \(B\), pointed straightening is a symmetric monoidal equivalence \[ \Fun (B,\An _*)\simeq (\An _{/B})_*, \] where the source carries the pointwise smash product and the target carries the smash product induced from the cartesian monoidal structure on \(\An _{/B}\).
Proof. The equivalence \[ \Ar (\Fun (B,\An )) \simeq \Fun (B,\Ar (\An )) \] identifies pushout products and cofibers on the left with their pointwise counterparts on the right. The symmetric monoidal localization of Lemma 16.3.4 therefore upgrades the canonical equivalence \(\Fun (B,\An )_*\simeq \Fun (B,\An _*)\) to a symmetric monoidal equivalence, where the right-hand side carries the pointwise smash product. Unpointed straightening \(\Fun (B,\An )\simeq \An _{/B}\) preserves finite products, since pointwise products correspond to fiber products over \(B\). Applying Lemma 10.5.1 finishes the proof. □
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