Colimit-preserving extension from presheaves

Theorem 1.8.9 (Lurie (2009), Theorem 5.1.5.6). Let \(C\) be a small \(\infty \)-category and let \(D\) be a large \(\infty \)-category admitting all small colimits. Then restriction along the Yoneda embedding \(Y\colon C \hookrightarrow \PSh (C)\) induces an equivalence \[ Y^*\colon \Fun ^{\colim }(\PSh (C),D) \quad \iso \quad \Fun (C,D), \] where the left-hand side denotes the full subcategory of colimit-preserving functors \(\PSh (C) \to D\).