Let $\mathbf P:=[{\mathbf{C}}^{\operatorname{op}},\mathbf{Set}]$ be the category of presheaves on $\mathbf C$. Let $*,\Omega:\mathbf C^{\operatorname{op}}\to \mathbf{Set}$ be, respectively, the constant functor on the singleton and the functor sending an object $c$ to the set of sieves on $c$. In $\mathbf P$ the subobject classifier is $t:*\to \Omega$, whose component $t_c$ sends $\{*\}$ to the maximal sieve on $c$.
I'm not sure on how to define the characteristic map. Let $X,Y\in \mathbf P$, and let $i:Y\hookrightarrow X$ be a monomorphism, that here is a natural transformation whose all components are injective. Consider the following square: $\require{AMScd}$ $$\begin{CD} Y(c)@>>> *\\ @VVi_cV @VVt_cV \\ X(c)@>\chi_c>> \Omega(c) \end{CD}$$ ($1$) in order to be cartesian, $\chi_c$ must send $x\in X(c)$ to the maximal sieve iff $x\in Y(c)$.
For any arrow $f:c'\to c$ in $\mathbf C$, if $\chi$ is natural this square commutes: $$\begin{CD} X(c)@>\chi_c>> \Omega(c)\\ @VVX(f)V @VV\Omega(f)V \\ X(c')@>\chi_{c'}>> \Omega(c') \end{CD}$$ here $\Omega(f)$ sends a sieve $T$ on $c$ to the sieve consisting of the arrows $g:c''\to c'$ such that $f\circ g\in T$. If $f\in T$, then $\Omega(f)(T)$ is the maximal sieve, clearly; but also the converse holds, i.e. if $\Omega(f)(T)$ is the maximal sieve (i.e. contains $\operatorname {id}_c$), necessarily $f\circ\operatorname{id}_c=f\in T$. Hence take $x\in X(c)$, and let $S:=\chi_c(x)$; $f\in S$ iff $\Omega(f)(S)$ is the maximal sieve, and using ($1$), iff $X(f)(x)\in Y(c')$. This should be indeed the definition of $\chi$.
Naturality of $\chi$: consider the second square and take $x\in X(c)$. Following the top-right route one gets the sieve consisting of the arrows $g:c''\to c'$ such that $f\circ g\in \chi_c(x)$; following the left-bottom route one gets the sieve of arrows $g:c''\to c'$ such that $X(g)(X(f)(x))\in Y(c'')$. By the definition of $\chi$, these sieves should coincide. The fact that the first square is cartesian, finally, is straightforward.
I'm sorry if this question may be a triviality, but the quantity of morphisms arising in these contexts confuse me abundantly, so I'd like to know if I have any misconception before continuing the theory. Thanks in advance.