On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any topos we have that the power object of an arbitrary object is (isomorphic to) its exponential with the subobject classifier.

It is also mentioned that in the case $c\cong{\bf 1}$ the power object becomes a subobject classifier. This is easy to see considering the diagram

in the case $c\cong{\bf 1}$ since ${\bf1}\times\Omega^{\bf1}\cong\Omega$ and ${\bf 1}\times d\cong d$, which means that $\in_{\bf 1}\cong{\bf 1}$ with the mono in question being 'true' $\top:{\bf 1}\to \Omega$, $\chi_m:d\to\Omega$ the characteristic function of $r$, and the top of the square being $!:r\to{\bf 1}$.

I'm trying to understand the object $\in_c$ when $c\ncong{\bf 1}$, both in ${\bf Sets}$ and more generally in any topos. I've tried 'fusing' the universal properties of an exponential and a subobject classifier to produce $\in_c$, but it is unclear how the domain of $\top$ changes to yield something besides ${\bf 1}$ -- it seems like the monomorphism $\in_c\hookrightarrow c\times\Omega^c$ is perhaps a currying of some sort?

For a more precise version of the question:

Let $\mathcal{C}$ be a closed category with finite limits and a subobject classifier. How can we define $\in_c$ for an arbitrary $c\in{\bf Ob}_\mathcal{C}$ using just the above structure, and what set is $\in_c$ in the case $\mathcal{C}={\bf Sets}$?