Is there a good reference that discusses weighted limits through the lens of the co-Yoneda embedding?
Recall that the limit of a functor $F:\mathcal{C}\to{\bf Set}$ is canonically given by the set $${\bf Hom}_{{\bf Set}^\mathcal{C}}(\Delta1,F)$$ together with the natural transformation $$\pi:\Delta{\bf Hom_{{Set}^\mathcal{C}}}(\Delta1,F)\Rightarrow F,$$ $$\pi_C:{\bf Hom_{{Set}^\mathcal{C}}}(\Delta1,F)\to F(C),$$ $$\alpha:\Delta1\Rightarrow F\mapsto\alpha_C(0).$$ More generally, for a functor $F:\mathcal{C}\to\mathcal{D}$ we have a canonical 'limit presheaf' $${\bf Hom}_{\mathcal{D}^\mathcal{C}}(\Delta-,F):\mathcal{D}^{op}\to{\bf Set}$$ where a limit for $F$ is precisely an object whose representable presheaf is isomorphic to this limit presheaf. Further, since $${\bf Hom}_{\mathcal{D}^\mathcal{C}}(\Delta D,F)\cong{\bf Hom}_{{\bf Set}^{\mathcal{C}}}(\Delta1,{\bf Hom}_\mathcal{D}(D,F(-)))$$ $$\alpha:\Delta D\Rightarrow F \longmapsto \alpha':\Delta1 \Rightarrow {\bf Hom}_\mathcal{D}(D,F(-))$$ $$\alpha'_C(0)=\alpha_C,$$ for all $D\in{\bf Ob}_\mathcal{D}$ we arrive at the 'explicitly conical' definition of limits; denoting Yoneda and co-Yoneda embeddings of a category $\mathcal{D}$ by $y_\mathcal{D}$ and $y^{co}_\mathcal{D}$ respectively, a limit of $F$ is an object $\varprojlim F$ in $\mathcal{D}$ together with a natural isomorphism $$\pi:y_\mathcal{D}(\varprojlim F)\Rightarrow{\bf Hom}_{{\bf Set}^\mathcal{C}}(\Delta1,y^{co}_\mathcal{D}(-)\circ F),$$ More generally, for a weight $W:\mathcal{C}\to{\bf Set}$ the weighted limit of $F$ is an object $\substack{\varprojlim \\ W} F$ together with a natural isomorphism $$\pi^W:y_\mathcal{D}(\varprojlim_W F)\Rightarrow {\bf Hom}_{{\bf Set}^\mathcal{C}}(W,y^{co}_\mathcal{D}(-)\circ F).$$ Most references I can find for weighted limits, e.g. the nlab, discuss weighted limits pointwise in terms of the objectwise limit of the Yoneda embedding postcomposed with the functor in question, but that characterization is less intuitive to me than the above one. Any pointers are appreciated.
