Let $1-\operatorname{Cat}^{\operatorname{St,cocmpl}} _{\operatorname{cont}}$ denote the ($\infty$-)category of stable cocomplete ($\infty$-)categories. The Ind-completion $\operatorname{Ind}(C_0)$ of an idempotent complete small stable ($\infty$-)category $C_0$ is an example of such a category. The category $1-\operatorname{Cat}^{\operatorname{St,cocmpl}} _{\operatorname{cont}}$ comes with a tensor product $\otimes$ and a duality between two categories $C$ and $D$ in $1-\operatorname{Cat}^{\operatorname{St,cocmpl}} _{\operatorname{cont}}$ is the datum of a pair of morphisms $$ \epsilon:C \otimes D \rightarrow Sp, \ \mu:D \otimes C \rightarrow Sp$$ such that the composition $(\epsilon \otimes id_C) \circ (id_C \otimes \mu)$ and $(id_D \otimes \epsilon) \circ (\mu \otimes id_D)$ are isomorphic to the identifies. We say $C$ and $D$ are dual to each other if there exists a duality between $C$ and $D$. Note such a duality is unique so one can say the dual of $C$ and denote it by $C^\vee$ if it exists.
A standard theorem is that if $C = \operatorname{Ind}(C_0)$ comes from Ind-completion, then $C$ is dualizable and its dual is given by $$ C^\vee = \operatorname{Ind}(C_0^{op}).$$ In this case, the map $\epsilon$ is induced from the natural pairing $$ C_0^{op} \times C_0 \rightarrow Sp, \ (c,d) \mapsto \operatorname{Map}_{C_0}(c,d)$$ by the universal properties of tensor product and Ind-completion. Now my question is the following: The existing proofs which I can find usually claim that it's enough to prove that there exists an equivalence: $$C^\vee \otimes B = \operatorname{Fun}^L(C,B)$$ which is functorial for $B \in 1-\operatorname{Cat}^{\operatorname{St,cocmpl}} _{\operatorname{cont}}$. Here $\operatorname{Fun}^L$ is the internal Hom in $1-\operatorname{Cat}^{\operatorname{St,cocmpl}} _{\operatorname{cont}}$. I can understand that this equivalence provide the map $$ \mu: Sp \rightarrow C^\vee \otimes C$$ by picking up the identity functor in $\operatorname{Fun}^L(C,C)$. But how can one use the above equivalence to check the requirement for duality?
This is a general fact about symmetric monoidal categories $\mathcal C$. For $C \in \mathcal C$, the internal hom functor $[C,-]: \mathcal C \to \mathcal C$ is (when it exists) the right adjoint of $C \otimes (-): \mathcal C \to \mathcal C$ (by definition). IF $C$ has a dual $C^\vee$, then $C^\vee$ is right adjoint to $C$ when viewed as a 1-morphism in the 1-object bicategory $B\mathcal C$ whose morphisms are the objects of $\mathcal C$.
The condition that you quote says that there is an object $C^\vee$ such that $C^\vee \otimes (-)$ is naturally isomorphic to $[C,-]$, i.e. that $C^\vee \otimes (-)$ is right adjoint to $C \otimes (-)$. Taking the components of the unit and counit natural transformations at the unit object of $\mathcal C$ yields a unit and counit exhibiting $C$ and $C^\vee$ as dual (this last argument should be basically the 2-Yoneda lemma, but I am having difficulty precisely stating the connection at the moment).