Strong monoidal transport from a product-base cocartesian family

Straighten $q$ to $F:B\times S\to {\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }}$. Currying is the exponential equivalence $\mathrm{Fun}(B\times S,{\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }})\simeq \mathrm{Fun}(B,\mathrm{Fun}(S,{\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }}))$, so it gives one functor $G:B\to \mathrm{Fun}(S,{\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }})$, with $G(b)=F(b,-)$. The pullback $E_b^{\otimes }\to S$ is its unstraightening. By the assumed inert Segal equivalences and terminal zeroth value, this is a symmetric monoidal infinity-category.

An edge $u:b\to b^{\mathrm{\prime }}$ gives the actual natural transformation $G(u):F(b,-)\to F(b^{\mathrm{\prime }},-)$ in $\mathrm{Fun}(S,{\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }})$. Under the straightening equivalence whose morphisms preserve cocartesian arrows, it gives a map $E_b^{\otimes }\to E_{b^{\mathrm{\prime }}}^{\otimes }$ OVER $S$ preserving ALL $S$-cocartesian arrows. Its component at $s\in S$ is the $q$-transport along $(u,{\mathrm{i}\mathrm{d}}_s)$. Thus it is strong symmetric monoidal by the criterion in [F1], not just a map preserving inert arrows.

Explicitly, for every $\alpha :s\to t$ in $S$, naturality in the curried functor gives an equivalence $F(b^{\mathrm{\prime }},\alpha )\circ F(u,{\mathrm{i}\mathrm{d}}_s)\simeq F(u,{\mathrm{i}\mathrm{d}}_t)\circ F(b,\alpha )$. Both sides are transports along the same composite arrow $(u,\alpha )$ in the product base. For $\alpha :⟨2⟩\to ⟨1⟩$ this is the tensor equivalence, and for $\alpha :⟨0⟩\to ⟨1⟩$, together with terminal $E_{(b,0)}$, it is the unit equivalence. Hence the active and nullary operations are preserved, not merely the coordinate projections.

Every simplex of $B$ is already sent by $G$ to a simplex of $\mathrm{Fun}(S,{\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }})$. Unstraightening its transformations retains their full compositions and all higher coherences. The full subcategory of $\mathrm{Fun}(S,{\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }})$ on objects satisfying the Segal/unit conditions contains every $G(b)$, and, by step 2.1, its transformations correspond to strong symmetric monoidal functors. Therefore $G$ factors through that model of symmetric monoidal infinity-categories, proving the asserted coherent diagram. No higher coherence has been inferred from pointwise equivalences.