I am asking this as I have established a functor F between categories C and D such that F is faithful, full and surjective on objects. Can I say that F is an equivalence of categories? I think so but need some expert comment for assurance. Thanks in advance.
2026-03-26 01:28:17.1774488497
Is it true that functors which are surjective on objects are obviously essentially surjective?
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Yes, it is obvious that a functor which is surjective on objects is also essentially surjective on objects. In your given situation, it is therefore okay to state that “the functor $F$ is faithful, full and surjective on objects, and therefore an equivalence of categories”.¹
¹ Assuming that it is known to the reader that a functor which is faithful, full and essentially surjective is an equivalence of categories.