I'm trying to read the article about spaces and quantities on ncat. I find it very interesting, but I'm afraid I can't follow how technically they are defining the functors – either because it's supposed to be an informal exposition, or because they are throwing around Yoneda embeddings way too liberally for me, or both!
Take a category $S$, and call $P:=\operatorname{Set}^{S^\text{op}}$ and $Q:=\operatorname{Set}^S$.
They claim that there is a pair of functors, $C: P^\text{op} \to Q$ and $\operatorname{Spec}: Q\to P^\text{op}$, that are adjoint.
But how are they defined? I.e. if I have a presheaf (on $S$) $X$, what is the copresheaf (on $S$) $C(X)$?
Sorry I know it's just written there. Any help would be really appreciated!