Proof.
Consider the chain of equivalences \[\mathcal C\otimes \mathcal P(J) \simeq \mathrm{Fun^L}(\mathcal P(J), \mathcal C^{\mathrm{op}})^{\mathrm{op}} \simeq \mathrm{Fun}(J, \mathcal C^{\mathrm{op}})^{\mathrm{op}} \simeq \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] Here, the first equivalence follows from ([002Z]), the second equivalence is the universal property of the Yoneda embedding [Lur09, Thm. 5.1.5.6], and the last records the interplay between functor categories and opposites. Precomposing this equivalence with the inclusion functor \(\mathcal C\times J \rightarrow\mathcal C\otimes \mathcal P(J)\) (which is cocontinuous in its second argument) unpacks to the functor ([0063]). ◻