Any compactly generated presentable stable $\infty$-category $C$ is known to be dualizable (with respect to Lurie's tensor product), so there is a coevaluation map:


$$Sp \to C \otimes C^{dual}.$$


Can one describe this map (or the image of the sphere spectrum) more concretely in terms of the compact generators of $C$?



Counit map for compactly generated categories
Recall that, for $A,B$ stable presentable $\infty$-categories, we can compute the tensor and internal hom using the equivalences $A\otimes B \simeq \operatorname{Fun}^\mathrm{lim}(A^\mathrm{op},B)$ and $\operatorname{hom}(A,B) \simeq \operatorname{Fun}^\mathrm{L}(A,B)$. If $A$ is moreover compactly generated, we also have $\operatorname{Fun}^\mathrm{L}(A,B) \simeq \operatorname{Fun}^\mathrm{ex}(A^\omega, B)$.


Thus, for $C$ a stable compactly-generated $\infty$-category, a colimit-preserving functor $\mathrm{Spt} \to C \otimes C^\vee$ is the same thing as an object of $C \otimes C^\vee \simeq \operatorname{Fun}^\mathrm{lim}(C^\mathrm{op}, \operatorname{Fun}^\mathrm{ex}(C^\omega,\mathrm{Spt}))$. So, what are some good functors $C^\mathrm{op} \times C^\omega \to\mathrm{Spt}$? One reliable choice is the (restricted) mapping spectrum functor $(x,y)\mapsto \operatorname{map}(x,y)$. This is a sensible choice, since the functor $C^\mathrm{op} \to \operatorname{Fun}(C^\omega,\mathrm{Spt})$ given by $x\mapsto \operatorname{map}(x,{-})$ preserves limits and factors through the full subcategory $\operatorname{Fun}^\mathrm{ex}(C^\omega,\mathrm{Spt})$ of the target. And indeed, one can check that this, together with the evaluation functor $C^\vee \otimes C \to \mathrm{Spt}$ defined by $(F, x) \mapsto F(x)$, satisfies the triangle identities for dual objects.


Alternatively, the object of $C\otimes C^\vee$ that we are interested in can be described as the coend $\int^{x\in C^\omega} x\otimes \operatorname{map}(x,{-})$, which is might align better with one's linear-algebraic intuitions.


