$\newcommand{\op}{\mathrm{op}}$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 $$y:1_{\bf Cat}\Rightarrow{\bf Set}^{-^{\op}}$$ whose components at each category are given by the various Yoneda embeddings, except that the appropriate dinaturality diagram
for a functor $F:\mathcal{C}\to\mathcal{D}$ doesn't commute on the nose, it only commutes up to $$F_{-,-}:{\bf Hom}_\mathcal{C}(-,-)\Rightarrow{\bf Hom}_\mathcal{D}(F(-),F(-))$$ where ${\bf Hom}_\mathcal{C}(-,-),{\bf Hom}_\mathcal{D}(F(-),F(-)):\mathcal{C}^{\op}\times\mathcal{C}\to{\bf Set}$ are the hom-functor at $\mathcal{C}$ and the hom-functor induced by $F$, respectively. We further have that $F$ 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, so
what are some references leveraging this view on Yoneda, or some interesting consequences of it?
Commuting 'up to a $2$-cell in $\mathfrak{Cat}$' makes me think this is probably clearest from a $2$-categorical perspective; we can extend the above discussion to be about ${\bf Set}^{-^{\op}}:\mathfrak{Cat}\to\mathfrak{Cat}^{(1,2)-\op}$, which is what a written-up reference probably does. Any pointers are appreciated.
