Let $F: C \to C'$ be a functor and let $A \in C'$.
The category $C_A$ is given by $\text{Ob}(C_A) = \{(X, s); X \in C, s : F(X) \to A \}$ and $\text{Hom}_{C_A}((X,s), (Y, t)) = \{ f \in \text{Hom}_C(X,Y); s = t\circ F(f)\}$.

But then after explaining Yoneda lemma and the convention $X(Y) \equiv \text{Hom}_C(Y, X)$ so that might be in play here.

Then on pg. 26 does this:

Let $F \in C^{\wedge}$. Then $F$ is representable if and only if $C_F$ has a terminal object. Proof. Let $(X, s) \in C_F$, that is, $X \in C$ and $s \in F(X)$. For any $(Y, t) \in C_F$, $\text{Hom}_{C_F} ((Y, t), (X, s)) \simeq \{u \in \text{Hom}_{C}(Y, X); F(u)(s) = t\}$. Hence, $(X, s)$ is a terminal object of $C_F$ if and only if $\text{Hom}_{C_F}((Y,t), (X, s)) \simeq \{\text{pt}\}$ for any $Y \in C$ and $t \in F(Y)$, and this condition is equivalent to saying that the map $\text{Hom}_C(Y, X) \to F(X)$ given by $u \to F(u)(s)$ is bijective for any $Y \in C$.

First of all, how are they defining $C_F$ for a functor $F$? That was never mentioned in the preceeding pages, mon.


Well, $C^{\land}$ is the category of functors $C^{op}\to Set$ as objects and natural transformations between them as arrows. And we consider the Yoneda embedding as our functor $y:C\to C^\land$.

Now $(X,s)$ being an object in $C_F$ means that $X$ is an object of $C$ and $s:y(X)\to F$, where $y(X)=\hom_C(-,X)$ and $s$ is a natural transformation, which can be identified with an element of the set $F(X)$ by the Yoneda lemma.


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