===
Let \(\mathcal{C}\) be a small category, \(\mathcal{A}\) a cocomplete category; then, precomposition with the Yoneda embedding \(y_{\mathcal{C}} : \mathcal{C} \to \widehat{\mathcal{C}}\) determines a functor
\[\textsf{Cat}(\widehat{\mathcal{C}}, \mathcal{A})\xrightarrow{\_\circ y_{\mathcal{C}}} \textsf{Cat}(\mathcal{C},\mathcal{A}).\]
- The universal property of the category \(\widehat{\mathcal{C}}\) amounts to the existence of a left adjoint \(L_y\) to the above functor, that has invertible unit (so, the left adjoint is fully faithful).
- The essential image of \(L_y\) consists of those \(F : \widehat{\mathcal{C}} \to \mathcal{A}\) that preserve all colimits.
- If \(\mathcal{A} = \widehat{\mathcal{D}}\), this essential image is equivalent to the subcategory of left adjoints \(F : \widehat{\mathcal{C}} \to \widehat{\mathcal{D}}\).