Isthe Yoneda embedding part of a larger structure? The Yoneda embedding for a category $A$ embeds it in its category of presheaves $\hat{A}:=Set^{A^{op}}$ as $y_A:A \rightarrow \hat{A}$. This makes it look as though it should be part of larger 'structure' - perhaps the component of some natural transformation, except here the arrow is between categories rather than functors. But if we use the identity functor for $Cat$, that is $1_{Cat}$, then we can rewrite the the first term as $A=1_{cat}A$ but the second term doesn't appear to re-writable in the same way.
So can we have a functor $y:Cat\rightarrow Cat$ such that $y(A)=y_A$ and for a functor $F:A\rightarrow B$, we have $y(F):y_A \rightarrow y_B$?
Is this even the right way to think about it?
 A: You can characterize the Yoneda functor $y\colon\mathcal C\to \widehat{\cal C}$ as a free cocompletion operation.
In a few words [and size issues apart: from now on everything happens in a universe of sets; maybe some of you can correct my lack of detail], you are considering a left adjoint to a functor which regards cocomplete categories as a sub(2-)category of all categories, which starts from $\cal C$ and enlarges it until it added all possible colimits for all possible shape of diagrams.
In a much more conceptual way, you are looking for something which gives you an equivalence of categories between $\mathbf{Funct}({\cal C,D})$ and $\mathbf{Cocont}({\widehat{\cal C},\cal D})$ for any cocomplete category $\cal D$; or in other words, you are looking for a procedure which exhibits a category $\widehat{\cal C}$ out of $\cal C$, such that any $F\colon\cal C\to $ {cocomplete} extends uniquely to a cocontinuous functor $\hat F\colon \widehat{\cal C}\to ${cocomplete}.
I can give you more details about how to build $\hat F$, but I think they could confuse you if you lack a couple of prerequisites; so tell me if you'd rather digest this, first :)
Another interesting point is that this is an "extremist" way to add colimits; you can add things little by little and yet continue to live happy! For example you can fix a prescribed class of limits, say $\mathcal J\subset\mathbf{Cat}$ and consider a functor $\mathsf{Ind}_\cal J$ which adds only $\cal J$-shaped colimit; for example if $\cal J$ contains only the empty category, your empty-completed $\cal C$ contains an initial object. If $\cal J$ contains the discrete category $\{0,1\}$, then $\mathsf{Ind}_\cal J(\cal C)$ has finite coproducts... 
This point of view opens another Pandora's box; I suggest you to google some keywords like "ind-completion", "presentable and accessible category".
A: Let $\mathbf{Cat}$ denote the category of small categories, $\mathbf{Cat}_{LS}$ denote the category of locally small categories(see Remark 1). There exists an obvious inclusion functor $V\colon \mathbf{Cat}\to\mathbf{Cat}_{LS}$(which sends $A$ to $A$, because every small category is locally small). Then denote by $D\colon\mathbf{Cat}\to\mathbf{Cat}$ the dualization functor, such that $D(A)=A^{op}$ and $D(T)=T^{op}$. Also we can construct the generalized inverse image functor $R\colon\mathbf{Cat}^{op}\to\mathbf{Cat}_{LS}$, such that $R(A)=\mathbf{Set}^{A}$ and $R(T)=T^*$(see Remark 2). Then we can consider a composition:
$$
\mathbf{Cat}^{op}\xrightarrow{D^{op}}\mathbf{Cat}^{op}\xrightarrow{R}\mathbf{Cat}_{LS}
$$
The functor $R\circ D^{op}$ sends $A$ to $\mathbf{Set}^{A^{op}}$, as you wanted. But we can't consider the Yoneda embedding as a natural transformation $V\to R\circ D^{op}$, because the first functor is from $\mathbf{Cat}$, but the second is from $\mathbf{Cat}^{op}$.
Remark 1. One can rightly note, that actually $\mathbf{Cat}_{LS}$ is not a category in the usual sense(because its objects don't form a class). But we can consider $\mathbf{Cat}_{LS}$ as a metacategory(see [MacLane, CWFM, 1 chapter] for the reference).
Remark 2. Let $A$ and $B$ be categories, $T\colon A\to B$ be a functor between them. Then we can define the inverse image functor $T^*\colon\mathbf{Set}^{B}\to\mathbf{Set}^A$, such that:
$$
T^*(S)=S\circ T;
$$
$$
(T^*(\alpha))(a)=\alpha(T(a)),
$$
for any $S\colon B\to\mathbf{Set}$ and any $\alpha\colon S\to S'$(where $S,S'\colon B\to\mathbf{Set}$). It is an easy exercise to check that $T^*(\alpha)$ is indeed a natural transformation and $T^*$ is indeed a functor.
