Tensoring by spectra is left adjoint to hom

Proof. The uniqueness of \(X \otimes Y\) is clear from the stable Yoneda lemma, Corollary 4.4.6. For existence, we need to show that for every \(Y \in C\) the functor \[ \Phi \colon \Sp \catop \to \Fun (C,\Sp ), \qquad \Phi (X) := \hom _{\Sp }(X,\hom _C(Y,-)) \] lands in the full subcategory \(C\catop \hookrightarrow \Fun (C,\Sp )\) of corepresentable functors, i.e. functors of the form \(\hom _C(W,-)\) for some \(W \in C\).

Step 1: We first show that \(\Phi (X)\) is corepresentable whenever \(X = \S [X']\) is an unreduced suspension spectrum for some anima \(X' \in \An \). To this end, recall from Theorem 1.8.9 that evaluation at \(*\) induces an equivalence \[ \Fun ^{\colim }(\An , C) \quad \iso \quad C. \] We let \(- \otimes Y\colon \An \to C\) denote the unique colimit-preserving functor satisfying \(* \otimes Y \cong Y\). We then claim that there exists a natural isomorphism \[ \Phi (\S [X']) \quad \cong \quad \hom _C(X' \otimes Y, -) \] of functors \(C \to \Sp \) for every \(X' \in \An \). For this, note that both sides define colimit-preserving functors \(\An \to \Fun (C,\Sp )\catop \) by varying \(X'\). Indeed, on either side a colimit in \(X'\) becomes a limit in \(\Fun (C,\Sp )\), because the relevant mapping-spectrum functor is contravariant in the variable determined by \(X'\); this is precisely a colimit after passing to \(\Fun (C,\Sp )\catop \). It therefore suffices to produce an isomorphism for \(X'= *\). And indeed, in that case both sides evaluate to \(\hom _C(Y,-)\) by Lemma 4.4.8 and the given isomorphism \(* \otimes Y \cong Y\).

Step 2: We now show the claim for an arbitrary spectrum \(X\). Observe that the functor \(\Phi \) is exact and sends colimits of spectra to limits in the functor category. Also observe that the corepresentable functors in \(\Fun (C,\Sp )\) are closed under limits and shifts. It thus remains to show that every spectrum \(X\) can be built from unreduced suspension spectra using shifts, cofibers and colimits. By Corollary 4.3.30, every spectrum is a colimit of shifts of reduced suspension spectra. Moreover, Remark 4.4.3 writes every reduced suspension spectrum \(\Sigma ^{\infty }(Z,z)\) as the cofiber of a map \(\S \to \S [Z]\) between unreduced suspension spectra. □