Corollary 5.3.24. Let \(A\) be a semiadditive \(\infty \)-category and let \(C\) be an \(\infty \)-category with finite products. The forgetful functor \(\CMon (C)\to C\) induces an equivalence \[ \Fun ^{\times }(A,\CMon (C))\iso \Fun ^{\times }(A,C). \]
Proof. By Yoneda, it suffices to show that this induces an equivalence on mapping animae out of every \(\infty \)-category \(E\). This follows from the natural equivalences \begin {align*} \Hom _{\Cat _{\infty }}\bigl (E,\Fun ^{\times }(A,\CMon (C))\bigr ) &\simeq \Hom _{\Cat ^{\mathrm {prod}}_{\infty }}\bigl (A,\Fun (E,\CMon (C))\bigr ) \\ &\simeq \Hom _{\Cat ^{\mathrm {prod}}_{\infty }}\bigl (A,\CMon (\Fun (E,C))\bigr ) \\ &\simeq \Hom _{\Cat ^{\mathrm {prod}}_{\infty }}\bigl (A,\Fun (E,C)\bigr ) \\ &\simeq \Hom _{\Cat _{\infty }}\bigl (E,\Fun ^{\times }(A,C)\bigr ), \end {align*}
where the second equivalence uses pointwise finite products and the third is the adjunction from Proposition 5.3.23. □
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