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free cocompletion

The Yoneda embedding \(\mathsf{C}\to \mathsf{P_C} \coloneqq \underset{ \mathsf{pre} } {\mathsf{Sh} }(\mathsf{C}, {\mathsf{Set}})\) has a universal property: for any complete category \(\mathsf{D}\), \({\mathsf{Fun}}(\mathsf{C}, \mathsf{D}) \simeq{\mathsf{Fun}}^L(\mathsf{P_C}, \mathsf{D})\) where the latter are colimit-preserving functors. This embedding is left exact and preserves limits.