Is the Yoneda bijection natural in the domain category?

Question

In the Yoneda Lemma, we start with a locally small category (the domain category, sometimes called a site), together with some other data, and establish a certain bijection. What is the naturality (or functorality) of the bijection under change of domain categories, first via a functor, then under a natural transformation between such functors? (Without going into the gory details here, are there some textbooks or papers that answer that question?)

Answer 1

$\require{AMScd}$I suspect this is not what you were asking, but it's the only way in which it makes sense to answer.

Let's say you have a functor $H : \mathcal C \to \mathcal D$. You can use it in the diagram $$\begin{CD} \mathcal C @>y>> [\mathcal{C}^\text{op},{\bf Set}] \\ @VHVV@. \\ \mathcal D @>>y> [\mathcal{D}^\text{op},{\bf Set}] \\ \end{CD}$$ and now you have two choices to fill this with a functor between the presheaf categories, because $H$ induces a pair of adjoint functors $$ H^* : [\mathcal D^\text{op},{\bf Set}] \rightleftarrows [\mathcal C^\text{op}, {\bf Set}] : H_* $$ $H^*$ precomposes $H$ (well, to be precise, the opposite functor $H^\text{op}$) to a functor $F : \mathcal D \to \bf Set$ in order to obtain a functor $FH : \mathcal C^\text{op} \to \bf Set$. $H_*$ acts in a more complicated way, it is a certain colimit built out of $H$ and a presheaf $G$ on $\mathcal C$.

Now, choosing $H^*$ to fill the diagram you don't get a naturality square, and not even a commutative diagram, because there is just a natural transformation filling it, which is invertible if and only if $H$ is fully faithful (try to prove it).

Choosing $H_*$ turns out to give you a (pseudo)commutative square, which is already better. But the definition of $H_*$ is quite elusive!