Denote by $\mathsf{CAT}$ the category of all locally small categories, by $\mathsf{Set}$ the category of sets, and write $ [\mathsf{C}, \mathsf{D}]$ for $\mathsf{CAT}(\mathsf{C}, \mathsf{D})$. Also, $\mathrm{op} \colon \mathsf{CAT} \to \mathsf{CAT}$ will be the automorphism $\mathsf{C} \mapsto \mathsf{C}^{\mathrm{op}}$.
I cite this fact, that is clear:
Let $P \colon \mathsf{CAT} \to \mathsf{CAT}$ be the functor defined by $\mathsf{C} \mapsto [\mathsf{C}^{\mathrm{op}}, \mathsf{Set}]$, and $Q \colon \mathsf{CAT} \to \mathsf{CAT}$ the functor $\mathsf{C} \mapsto [\mathsf{C}, \mathsf{Set}]^{\mathrm{op}}$. There is a commutative square of functors $$ \require{AMScd} \begin{CD} \mathsf{CAT} @>{P}>> \mathsf{CAT} \\ @VV\mathrm{op}V @VV\mathrm{op}V \\ \mathsf{CAT} @>{Q}>> \mathsf{CAT}. \end{CD} $$
What I don't understand are two statements following from the observation above:
there are two Yoneda embeddings $\mathsf{C} \mapsto P\mathsf{C}$ and $\mathsf{C} \mapsto Q\mathsf{C}$. What does this mean? The (contravariant) Yoneda embedding of $\mathsf{Set}$ should be $[-, \mathsf{Set}]$, that is different from both $P = [-^{\mathrm{op}},\mathsf{Set}]$ and $Q = [-, \mathsf{Set}]^{\mathrm{op}}$;
as a consequence of the commutativity of the square of functors above, representables switch; i.e., a contravariant functor $\mathsf{C} \to \mathsf{Set}$ defined by $C \mapsto \mathsf{C}(-, C)$ is the same as the functor $\mathsf{C}^{\mathrm{op}} \to \mathsf{Set}$ defined by $C \mapsto \mathsf{C}^{\mathrm{op}}(C, -)$. Now, the second is a functor in $P\mathsf{Set}$, but the first is not an object of $Q\mathsf{Set}$, because there is no relation between $[\mathsf{C}, \mathsf{Set}]^{\mathrm{op}}$ and contravariant functors $\sf{C} \to \mathsf{Set}$. And even if the objects of $Q\mathsf{Set}$ were such contravariant functors, I should have a map $P\mathsf{Set}\to Q\mathsf{Set}$ to put in relations functors $\mathsf{C}^{\mathrm{op}} \to \mathsf{Set}$ and contravariant functors $\mathsf{C} \to \mathsf{Set}$; so I'd need more a natural transformation $P \Rightarrow Q$ than a diagram of functors. (I understand that representables switch, I just don't see how it is a consequence of the indented paragraph).
Would someone help me to get a grasp on this? Thanks.