For a locally small category $\mathcal{C}$, let $y_\mathcal{C}:\mathcal{C}\to{\bf Set}^{\mathcal{C}^{op}}$ denote the Yoneda embedding at $\mathcal{C}$. Letting ${\bf Cat}$ denote the $1$-category of locally small categories, we have a functor $${\bf Set}^{-^{op}}:{\bf Cat}\to{\bf Cat}^{op}$$ $$\mathcal{C}\mapsto{\bf Set}^{\mathcal{C}^{op}}$$ $$F:\mathcal{C}\to\mathcal{C}'\longmapsto\circ F^{op}:{\bf Set}^{\mathcal{C}'^{op}}\to{\bf Set}^{\mathcal{C}^{op}}$$ where $\circ F^{op}$ is precomposition with the opposite functor of $F$, and the Yoneda embeddings together yield a dinatural transformation $$y:1_{\bf Cat}\Rightarrow{\bf Set}^{-^{op}}$$ whose components at each category are given by the various Yoneda embeddings. This is certainly common knowledge, leading to my question:

What are some references leveraging this view on Yoneda, or some interesting consequences of it?