One possible further addition to the list would be a closed graph theorem in the spirit of Theorem 6 for the category of finite etale covers of a fixed variety (without any need to assume normality or projectiveness, though I am not sure whether one can drop characteristic zero), basically because the degree argument used in the proof of Theorem 6 should also work in this category (though I am not 100% certain of this…). This should also be morally equivalent to the closed graph theorem for finite G-sets, where G is the etale fundamental group.
[ADDED LATER: it seems that in order for this to work, it is not enough for the graph of the putative morphism to be Zariski-closed; it must also be a finite etale cover of the base variety.]

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