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Moving the following old query out of the entry to here. Maybe it inspires somebody to add to the entry a remark towards the answer:
[ begin forwarded query ]
+– {: .query} Anonymous: Under what conditions are all injections in a category monomorphisms? Obviously injections are monomorphisms in a well-pointed topos or pretopos (those are models of particular types of set theories), but does that remain true in a (pre)topos without well-pointedness, a coherent category or an exact category?
Anonymous: There is this stackexchange post, but the answers only refer to concrete categories with a forgetful functor to Set and a free functor from Set, rather than arbitrary abstract categories. =–
[ end forwarded query ]
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