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Feb 6, 2019 at 22:04 comment added Mike Shulman Regarding terminology, MO points me to the following related question: mathoverflow.net/q/256382/49 which is about "Galois-initial" objects minus uniqueness of $h$ (which then requires specifying explicitly that it is an isomorphism).
Feb 6, 2019 at 20:58 comment added Mike Shulman I wouldn't be surprised if essentially this situation, or something very like it, has already been considered in categorical Galois theory, but I'm not sufficiently familiar with that literature to point to where.
Feb 6, 2019 at 20:56 history answered Mike Shulman CC BY-SA 4.0