Twisted arrow hom lax monoidal

Proof. The twisted arrow construction preserves finite products, hence preserves commutative monoid objects of \(\Cat _{\infty }\). Thus the symmetric monoidal structure on \(D\) induces a symmetric monoidal structure on \(\Tw (D)\). The source and target projections \[ s\colon \Tw (D) \to D\catop , \qquad t\colon \Tw (D) \to D \] are natural in \(D\), and therefore assemble to a symmetric monoidal functor \[ (s,t)\colon \Tw (D) \to D\catop \times D. \] We claim that the induced functor on total categories \((s,t)^{\otimes }\colon \Tw (D)^{\otimes } \to (D\catop \times D)^{\otimes }\) is still a left fibration. Under straightening over \(\Span (\Fin )\), this functor is induced by the natural transformation whose value on a finite set \(I\) is the \(I\)-fold product \[ (s,t)^I\colon \Tw (D)^I \to (D\catop \times D)^I \] of the usual source-target left fibration. Since left fibrations are stable under finite products, Lemma 23.2.2 shows that \((s,t)^{\otimes }\) is a left fibration. The fiber over an object \(\{(x_i,y_i)\}_{i \in I}\) is the anima \(\prod _{i \in I}\Hom _D(x_i,y_i)\), with transport induced by the symmetric monoidal structure on \(D\) and composition in \(D\).

Straightening this left fibration gives a functor \[ H^{\otimes }\colon (D\catop \times D)^{\otimes } \to \An . \] On the fiber over \(\lra {1}\) this is the usual Hom-functor \(\Hom _D(-,-)\colon D\catop \times D \to \An \). Moreover, the description of its fibers shows that \(H^{\otimes }\) preserves finite products: products in \((D\catop \times D)^{\otimes }\) are given by concatenating finite tuples, and \(H^{\otimes }\) sends such a tuple to the corresponding product of hom animae. By the universal property of the cartesian monoidal structure on \(\An \) from Theorem 15.3.11, this finite-product-preserving functor is precisely a lax symmetric monoidal structure on \(\Hom _D(-,-)\). □