The multiplicative Yoneda embedding

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 \(\Hom _D(-,-)\). □

Proof. By the universal property of Day convolution, a lax symmetric monoidal structure on \(Y\) is equivalent to a lax symmetric monoidal structure on the associated functor \[ D\catop \times D \to \An , \qquad (d',d) \mapsto \Hom _D(d',d). \] This is provided by Lemma 18.2.1.

It remains to show that this lax structure is strong. Let \(\{d_i\}_{i \in I}\) be a finite collection of objects of \(D\). We must show that the induced map \[ \bigotimes ^{\Day }_{i \in I} Y(d_i) \to Y\left (\bigotimes _{i \in I} d_i\right ) \] is an isomorphism in \(\PSh (D)\). Testing against an arbitrary presheaf \(H\colon D\catop \to \An \), the left-hand side represents natural transformations \[ \Nat \left (\prod _{i \in I} Y(d_i)(-), H\left (\bigotimes _{i \in I} -\right )\right ) \] of functors \((D^I)\catop \to \An \) by the defining property of the Day convolution tensor product. Applying the Yoneda lemma for \(D^I\) identifies this anima with \(H(\bigotimes _{i \in I} d_i)\), which is also \(\Nat (Y(\bigotimes _{i \in I} d_i),H)\). This proves the claim. □

Proof. Let \(x_1,\dots ,x_n,y\) be colors of \(\Oo \). Since \(Y\) is strong symmetric monoidal by Proposition 18.2.2, we have natural equivalences \begin {align*} \Mm _{\PSh (D)}(Yx_1,\dots ,Yx_n;Yy) &\simeq \Hom _{\PSh (D)}(Yx_1 \otimes ^{\Day } \dots \otimes ^{\Day } Yx_n,Yy) \\ &\simeq \Hom _{\PSh (D)}(Y(x_1 \otimes \dots \otimes x_n),Yy) \\ &\simeq \Hom _D(x_1 \otimes \dots \otimes x_n,y) \\ &\simeq \Oo (x_1,\dots ,x_n;y). \end {align*}

The first and last equivalences are induced by the unit \(\eta _{\Oo }\) and agree with the identifications of Lemma 17.3.17. This proves full faithfulness. For the final claim, let \(y \to F\) be an operadic limit cone in \(\Oo _{\lra {1}}\). By Remark 18.1.2, its image in \(D = \Env (\Oo )\) is an ordinary limit cone. Since the Yoneda embedding preserves limits, its image in \(\PSh (D)\) is again a limit cone. □