Proposition 22.1.2 (Lurie (2009), Proposition 5.3.6.2). Let \(\Kk \) be a collection of small \(\infty \)-categories. For every small \(\infty \)-category \(C\) and every (possibly large) \(\infty \)-category \(D\) with \(I\)-indexed colimits for all \(I \in \Kk \), restriction along the Yoneda embedding \(Y\colon C \hookrightarrow \PSh ^{\Kk }(C)\) induces an equivalence \[ \Fun _{\Kk }(\PSh ^{\Kk }(C),D) \iso \Fun (C,D), \] where the left-hand side denotes the full subcategory spanned by the functors \(\PSh ^{\Kk }(C) \to D\) that preserve \(I\)-indexed colimits for all \(I \in \Kk \).
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