Lemma 16.5.14. The evaluation functor \[ \ev _{S^0}\colon \Exc _*(\An _*^{\fin },\An _*) \to \An _* \] admits a left adjoint. It sends a pointed anima \(X\) to the excisive approximation of the smash product functor \[ P_1(X \wedge -) \colon \An _*^{\fin } \to \An _*. \]
Proof. First consider evaluation at \(S^0\) on the full functor category. Its left adjoint is the left Kan extension along \(S^0\colon *\to \An _*^{\fin }\), which sends a pointed anima \(X\) to the functor \[ K\longmapsto X\wedge \fgt (K)_+. \] Its value at the zero object is \(X\). Applying the reduction functor gives \(K\mapsto X\wedge K\), since the cofiber of the canonical map \(S^0\to \fgt (K)_+\) is \(K\). Thus \(X\wedge -\) is left adjoint to evaluation on reduced functors. Composing with the excisive approximation \(P_1\), which exists by Corollary 16.5.13, gives the asserted left adjoint on reduced excisive functors. □
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