Proposition 16.2.13. Let \(C\) be a small symmetric monoidal \(\infty \)-category and let \(D\) be a complete closed symmetric monoidal \(\infty \)-category for which the Day convolution symmetric monoidal structure on \(\Fun (C,D)\) exists. Then this symmetric monoidal structure is closed. Its internal hom is pointwise given by the end formula \[ \iHom _{\Fun (C,D)}(F,G)(X) \simeq \int _{Y \in C} \iHom _D(F(Y), G(X \otimes _C Y)). \] Here the integral denotes the end of Definition 23.6.3.
Proof. It suffices to show that for every third functor \(H\colon C \to D\) there is a natural equivalence \[ \Nat (H \otimes _{\Day } F, G) \simeq \Nat \left (H, \int _{Y \in C} \iHom _D(F(Y), G(- \otimes _C Y))\right ). \] Since the Day convolution is a left Kan extension, the left-hand side is equivalent to the anima \[ \Nat (H(-) \otimes _D F(-), G(- \otimes _C -)) \] of natural transformations of functors \(C \times C \to D\). By Proposition 23.6.6, this anima is the end \[ \int _{(X,Y) \in C \times C} \Hom _D(H(X) \otimes _D F(Y), G(X \otimes _C Y)). \] Using the Fubini rule for ends, Lemma 23.6.5, the adjunction \(- \otimes _D F(Y) \dashv \iHom _D(F(Y),-)\), and the fact that \(\Hom _D(H(X),-)\) preserves limits, we may rewrite this as \begin {align*} \int _{X \in C} &\int _{Y \in C} \Hom _D(H(X), \iHom _D(F(Y), G(X \otimes _C Y))) \\ &\simeq \int _{X \in C} \Hom _D\left (H(X), \int _{Y \in C} \iHom _D(F(Y), G(X \otimes _C Y))\right ). \end {align*}
Using the end description of animae of natural transformations once more gives the desired result. □
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