I'm trying to understand why the category of presheaves on a small category $\mathbf C$, the functor category $[\mathbf C^{\mathrm{op}},\mathbf{Sets}]$, is an elementary topos. Right now I need to find a classifier, and to do so I'm reading Section 4 of the first chapter of Sheaves in Geometry and Logic, the one called Typical Subobject Classifiers.
Here they explain how to define a classifier introducing sieves, but I need to do so without using them. I have to use, for each object $C$ of the category $\mathbf C$, the subfunctors of $\mathrm{Hom}_{\mathbf C}(-,C)$ instead of sieves on $C$. Now I look for a classifier $1\stackrel{\mathrm{true}}\longrightarrow\Omega$.
The functor $$1\colon \mathbf C^{\mathrm{op}}\rightarrow\mathbf{Sets},\qquad C\mapsto \{*\}$$ is a terminal object for $[\mathbf C^{\mathrm{op}},\mathbf{Sets}]$.
I define the functor $\Omega\colon \mathbf C^{\mathrm{op}}\rightarrow\mathbf{Sets}$ like this: for each object $C$ of $\mathbf C$, let $$\Omega(C)=\mathrm{Sub}_{[\mathbf C^{\mathrm{op}},\mathbf{Sets}]}\big(\mathrm{Hom}_{\mathbf C}(-,C)\big)$$ be the set of subfunctors of $\mathrm{Hom}_{\mathbf C}(-,C)$; and for each arrow $C'\stackrel f\to C$, let $$\Omega(f):\Omega(C)\rightarrow\Omega(C'),\qquad P\mapsto\Omega(f)(P),$$ where $\Omega(f)(P)$ makes the diagram $$\require{AMScd} \begin{CD} \Omega(f)(P) @>>> P \\ @V V V @VVV\\ \mathrm{Hom}_\mathbf C(-,C') @>>{f\circ-}> \mathrm{Hom}_\mathbf C(-,C) \end{CD} $$ into a pullback.
I define the natural transformation $1\stackrel{\mathrm{true}}\longrightarrow\Omega$ by $$\mathrm{true}_C\colon 1(C)=\{*\}\longrightarrow\Omega(C),\qquad *\mapsto\mathrm{Hom}_\mathbf C(-,C)$$ for all $C$ in $\mathbf C$.
And now, if I'm not wrong, I hope to prove that $1\stackrel{\mathrm{true}}\longrightarrow\Omega$ is a classifier for $[\mathbf C^{\mathrm{op}}, \mathbf{Sets}]$. I take a monic $Q\to P$ in $[\mathbf C^{\mathrm{op}}, \mathbf{Sets}]$ and I need to prove that there exists a unique natural transformation $\varphi$ that forms a pullback like this one: $$\require{AMScd} \begin{CD} Q @>>> 1 \\ @V V V @VV\mathrm{true}V\\ P @>>\varphi> \Omega \end{CD} $$
For every object $C$ of $[\mathbf C^{\mathrm{op}}, \mathbf{Sets}]$ and every element $x\in PC$, i know that this $\varphi$ is a natural transformation such that $\varphi_C(x)$ is a certain subfunctor of $\mathrm{Hom}_\mathbf C(-,C)$, and I think that, for every object $B$ in $\mathbf C$, this subfunctor gives $$\big(\varphi_C(x)\big)(B)=\{g\mid g\colon B\to C,\ (Pg)(x)\in QB\}.$$
Now I need to understand why this $\varphi$ is the only one which makes the previous diagram into a pullback. I cannot follow the proof of the book at this point. At page 39 they take a random natural transformation $\theta$ that makes the diagram into a pullback, and they prove that it needs to be equal to $\varphi$.
Given $x\in PC$ and an arrow $f\colon A\to C$, I can understand that $(Pf)(x)\in QA$ if and only if $$(\Omega f)(\theta_A x)=\text{true}_A(*),$$ and this means that the diagram $$\require{AMScd} \begin{CD} \mathrm{Hom}_{\mathbf C}(-,A) @>>> \theta_Cx \\ @V V V @VVV\\ \mathrm{Hom}_\mathbf C(-,A) @>>{f\circ-}> \mathrm{Hom}_\mathbf C(-,C) \end{CD} $$ is a pullback. But, from this point, how can I deduce that $\theta=\varphi$?