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 almost yield a [dinatural transformation](https://ncatlab.org/nlab/show/dinatural+transformation) $$y:1_{\bf Cat}\Rightarrow{\bf Set}^{-^{op}}$$ whose components at each category are given by the various Yoneda embeddings, except that the appropriate dinaturality diagram 

[![][1]][1]

for a functor $F:\mathcal{C}\to\mathcal{D}$ doesn't commute on the nose. To see what it does commute up to, recall that for any category $\mathcal{C}$ we have the 'hom-functor' ${\bf Hom}_\mathcal{C}(-,-):\mathcal{C}^{op}\times\mathcal{C}\to{\bf Set}$ sending pairs of objects to the hom-sets between them and pairs of arrows to the induced pre/postcomposition functions. Given a functor $F:\mathcal{C}\to\mathcal{D}$, we have an induced natural transformation $$F_{-,-}:{\bf Hom}_\mathcal{C}(-,-)\Rightarrow{\bf Hom}_\mathcal{D}\big(F(-),F(-)\big)$$

The dinaturality diagram above for an arbitrary functor $F:\mathcal{C}\to\mathcal{D}$ commutes up to $F_{-,-}$, and $F$ is fully faithful iff $F_{-,-}$ is a natural isomorphism, so the dinaturality diagram above commutes 'up to canonical iso' if we restrict our attention to the wide subcategory of ${\bf Cat}$ consisting of fully faithful functors.

This is certainly common knowledge, leading to my question:

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


  [1]: https://i.sstatic.net/CTUll.png