representable functor as kappa compact object

Let \(\mathcal{C}\) be an infinity category, \(\mathcal{P}(\mathcal{C})\) the infinity presheaf category and \(\mathrm{map}_{\mathcal{C}}(-,c)\) a presheaf for \(c \in \mathcal{C}\)
Then \(\mathrm{map}_{\mathcal{C}}(-,c)\) is a \(kappa\)-compact object for arbitrary \(\kappa\)

follows from the contravariant infinity yoneda lemma and pointwise calculation of colimits in a functor category

\begin{align*} \mathrm{map}_{\mathcal{P}(\mathcal{C})}(\mathrm{map}_{\mathcal{C}}(-,c), \mathrm{colim} \mathcal{G}) \cong& (\mathrm{colim} \mathcal{G})(C) \\ \cong& \mathrm{colim} \mathcal{G}(C) \\ \cong& \mathrm{colim} \mathrm{map}_{\mathcal{P}(\mathcal{C})}(\mathrm{map}_{\mathcal{C}}(-,c), \mathcal{G}) \end{align*}