subobject classifier of presheaf categories without using sieve As part of Leinster's Basic Category Theory exercise, I am trying to find a subobject classifier of presheaf category. Let $\textbf{A}$ be a small category. Then there is a functor $\text{Sub}:[\textbf{A}^{op},\textbf{Set}]^{op}\to\textbf{Set}$ sending every presheaf $X$ on $\textbf{A}$ to the set of subobjects of $X$. Here is the definition he used for the exercise:

A subobject classifier of $[\textbf{A}^{op},\textbf{Set}]$ is a representing object of $\text{Sub}$, provided it is representble

I have successfully use this to find a candidate of the subobject classifier $\Omega$, by assuming that it exists:
$$\text{Sub}(H_A)\cong H_\Omega(H_A)\cong [\textbf{A}^{op},\textbf{Set}](H_A,\Omega)\cong \Omega(A)$$
Here the first isomorphism is from the representability and the last one is from Yoneda Lemma.
Using this observation, we define $\Omega:\textbf{A}^{op}\to\textbf{Set}$ by
$$\Omega(A)=\text{Sub}(H_A)$$
$$\Omega(f:A'\to A)=\text{Sub}(H_f)$$
I would like to find a natural isomorphism $\alpha:\text{Sub}\to H_\Omega$. Here is where I got stuck: how do we define the component $\alpha_X$? A subobject of $X$ is a natural transformation into $X$, so an element in $\text{Sub}(X)$ has $X$ being the codomain. On the other hand, an element in $H_\Omega(X)=[\textbf{A}^{op},\textbf{Set}](X,\Omega)$ has $X$ being the domain
I have been searching around, but it seems like every explanation use the concept of sieve and subfunctors (which are something I haven't touched at all). I hope someone can give some insight and finish the argument
P.S. throughout this post, I use $H_\bullet$ for the Yoneda embeddings. The domains are made implicit for simplicity
 A: Here's the idea.
The monomorphisms in the presheaf category are pointwise monomorphisms, so we can identify subobjects of a presheaf $X\in[\mathbf{A}^\text{op},\newcommand\Set{\mathbf{Set}}\Set]$ with subpresheaves of $X$, in the sense of presheaves $F$ such that $F(a)\subseteq X(a)$ for all objects $a$, and for $f:a\to a'$, $X(f)$ sends elements of $F(a')$ to $F(a)$.
Now suppose we have a subpresheaf $F$ of some presheaf $X$. We want to construct a natural transformation $X\to \newcommand\Sub{\operatorname{Sub}}\Sub(H_{-})$.
Thus for $a\in \newcommand\A{\mathbf{A}}\A$, $\alpha\in X(a)$, we need to construct a subpresheaf of $H_a$. By Yoneda, $\alpha$ corresponds to a natural transformation
$H_a\to X$, so we can just take the preimage of $F$ in $H_a$. In other words, define $G_\alpha\newcommand\into\hookrightarrow\into H_a$ by
$$G_\alpha(a') = \{ f : a'\to a \text{ such that } f^*\alpha \in F(a')\subseteq X(a')\}.$$
Then we define $\eta : X\to \Sub(H_-)$ by $\eta \alpha = G_\alpha$.
Conversely, from a natural transformation, $\eta : X\to \Sub(H_-)$, we can recover the subobject $F$ by
$$F(a) =\{ \alpha\in X(a) \text{ such that } 1_a \in (\eta_a\alpha)(a)\subseteq H_a(a)\}.$$
Side note: a subfunctor is a subobject of a functor, and a sieve is a subobject of a representable functor, but we don't really need to use these words to prove the claim.
Edit:
To see that $\eta$ is natural, let $f:a\to a'$, let $\alpha\in X(a')$.
We need to show that $\eta f^*\alpha = f^*\eta\alpha$.
Now
$$
(\eta f^*\alpha)(a'')
= 
\{
g: a''\to a \text{ such that } g^*f^*\alpha \in F(a'')
\},
$$
and
$$
(f^*\eta\alpha)(a'')
=
(f_*)^{-1}((\eta\alpha)(a''))
=
\{
g: a''\to a \text{ such that } (f\circ g)^* \alpha \in F(a'')
\}.
$$
Thus, since $(f\circ g)^* = g^*f^*$, we have naturality.
Edit 2
I've been asked how we show that if we start with a natural transformation $\eta : X\to \Sub(H_-)$ and construct the associated subobject $F$ of $X$ how we show that the natural transformation $\overline{F}$ associated to $F$ is in fact $\eta$.
Let $a,a'\in \mathbf{A}$. Recall that
$$F(a) = \{ \alpha \in X(a) \text{ such that } 1_a \in \eta_a(\alpha)(a) \}.$$
We also know that if $\alpha \in X(a)$, then
$$
\overline{F}_a(\alpha)(a') 
= 
\{
g:a'\to a
\text{ such that }
g^*\alpha \in F(a)
\}.
$$
Putting these together we can compute
$$
\begin{aligned}
\overline{F}_a(\alpha)(a') 
&=
\{
g:a'\to a
\text{ such that }
1_{a'}\in \eta_{a'}(g^*\alpha)(a')
\}
\\
&=
\{
g:a'\to a
\text{ such that }
1_{a'}\in g^*(\eta_{a}\alpha)(a')
\}
\\
&=
\{
g:a'\to a
\text{ such that }
1_{a'}\in (g_*)^{-1}(\eta_{a}\alpha)(a')
\}
\\
&=
\{
g:a'\to a
\text{ such that }
g\circ 1_{a'}\in (\eta_{a}\alpha)(a')
\}
\\
&=
\{
g:a'\to a
\text{ such that }
g\in (\eta_{a}\alpha)(a')
\}
\\
&=(\eta_a\alpha)(a').
\end{aligned}
$$
Thus as subobjects of $H_a$, we have that
$\eta_a\alpha = \overline{F}_a\alpha$, as desired.
