Proposition 18.2.2. Let \(D\) be a small symmetric monoidal \(\infty \)-category. Then the Yoneda embedding \[ Y\colon D \to \PSh (D) = \Fun (D\catop ,\An ) \] admits a canonical strong symmetric monoidal structure, where \(\PSh (D)\) is equipped with the Day convolution monoidal structure.
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. □
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