I'm reading the page 54 of this pdf but don't understand the definition of characteristic arrow in $\text{Set}^{\mathbb{C}^{\text{op}}}$.
Assume that $F, G \in \text{Set}^{\mathbb{C}^{\text{op}}}$, $f:F\rightarrow G$. The characteristic arrow $\chi_{f}$ is defined as $\chi_{f}(A)(x):=\{f:B\rightarrow A|G(f)(x)\in G(B)\}$ for $A\in \mathbb{C}^{\text{op}}, x\in F(A)$.
I think "$x\in F(A)$" is typo of "$x\in G(A)$" because $\chi_{f}(A):G(A)\rightarrow \Omega (A)$ . However, then $G(g)(x)\in G(B)$ is trivial for all $g:B\rightarrow A$.
Would you tell me the precise definition of $\chi_{f}$?
There seem to be multiple typos. The correct definition is $$\chi_f (A) (x) = \left\{ h : B \to A \middle| \exists y \in F (B) . G (h) (x) = f_B (y) \right\}$$ or, in other words, $\chi_f (A) (x)$ is the sieve consisting of all $h : B \to A$ such that $G (h) (x)$ is in the image of $f : F \to G$.