Power object of terminal object in finitely complete category is claimed to be subobject-classifier in https://ncatlab.org/nlab/show/power+object. The connection is otherwise quite evident, but how to show $\in_1$ is necessarily terminal?
2026-03-28 10:17:21.1774693041
Power object of terminal object is subobject-classifier?
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Hint: For each object $X$, use the power object condition of $\Omega^1$ for the identity subobject $1_X\colon X\longrightarrow X$.