Replacing $\mathbf{Set}$ in Yoneda The Yoneda lemma (or rather the existence of the Yoneda embedding) states, roughly, that for each category $C$ there's an embedding (a fully faithful functor) of $C$ into $\mathbf{Set}^{C^{op}}$.
Which other categories $D$ have the property that for each category $C$, there's an embedding from $C$ into $D^{C^{op}}$?
In particular, is this true for each topos $D$? I guess one slogan of topos theory is that a topos is just as good as the category of sets, so I think it's reasonably to ask whether we can replace $\mathbf{Set}$ in the Yoneda lemma by any topod $D$.
 A: This is defenitly not an awnser, since it is full of vagueries, half truths, and bad analogies. But the comment section was to small to fit this remarks.
There is not always a functor $C\to D^{C^{op}}$, for this is equivalent as having a functor $C^{op}\times C \to D$. Enriched categories over $D$ provide such a functor. A category is a $\text{Set}$-enriched category, the functor $C^{op}\times C \to \text{Set}$ is given by the $\text{Hom}$ sets of your category $C$. Instead you always have a natural functor in $C$, $C \to D^{(D^C)}$ which takes an object $x$ of $C$ to the evaluation at $x$.
Note that the Yoneda lemma is very often stated as the fact that the evaluation functor at $x$ is represented by the functor represented by $x$, and then the functor $C \to \text{Set}^{(\text{Set}^C)}$ is given by composing $C \to (Set^{C})^{op}$ with $(Set^{C})^{op} \to \text{Set}^{(\text{Set}^C)}$
This parallels quite faithfully what happens in $k$-vector spaces : if you have an inner product on a vector space $V$, i.e. a map $V \otimes V \to k$, you get a linear map $V \to V^*$, and having such a linear map gives you an inner product. An inner product is not degenerated if this map is injective, giving some kind of Yoneda vibe to the situation.
Replacing $k$ with $\text{Set}$, you can think of categories as some kind of 'modules' over the category $\text{Set}$, having an 'inner product' (the $\text{Hom}$ spaces) that is 'not degenerated' (Yoneda embedding).
But you always have a natural linear map $V \to V^{**}$ that sends a vector $v$ to the linear map which evaluates a linear form on $V$ at the vector $v$.
