The definition for the partial map classifier $\eta_{b} : b \to b_{\bot}$I was given is that:
let $b$ be a member of a topos, and let $$\{\}_{b} : b \to \Omega^{b}$$ be the exponential adjoint of $$\delta_{b} : b \times b \to \Omega$$ (the exponential adjoint of $f: c \times d \to e$ is the unique arrow $g: c \to e^{d}$ s.t. $ev \circ g \times id_{d}= f$, and $\delta_{b}$ is the character of $\langle id_{b}, id_{b} \rangle : b \to b \times b$), and let $$h:\Omega^{b} \times b \to \Omega$$ be the character of $\langle \{\}_{b}, id_{b} \rangle : b \to \Omega^{b} \times b$, then let $$\hat{h}: \Omega^{b} \to \Omega^{b}$$ be the exponential adjoint of $h$. Finally, let $$f: b_{\bot} \to \Omega^{b}$$ be the equalizer of $id_{\Omega^{b}}$ and $\hat{h}$ then $\{\}_{b} = \hat{h} \circ \{\}_{b}$ so the unique arrow $$\eta_{b}: b \to b_{\bot}$$ s.t. $\{\}_{b} = f \circ \eta_{b}$ is the partial map classifier.
The problem is to prove that $\eta_{1}: 1 \to 1_{\bot}$ is the subobject classifier in the topos. The book I'm using is Goldblatt's Topoi book, which doesn't cover Yoneda's lemma and other useful tools, and perhaps because of that I'm consistently having a lot of trouble with these exercises. I'll greatly appreciate any answer, but perhaps someone can shed a light on this using only what's already been covered so far in his book?