Does it hold in every (concrete) category?

for the category of set: ok but that's the only proof we usually find on internet.

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Even in the category of sets, left-invertible is equivalent to injective only if the domain of the map is nonempty. And, still in the category of sets, the equivalence between surjectivity and right invertibility depends on (and in fact is equivalent to) the axiom of choice. Since counterexamples in other categories are well-known, this question seems more appropriate for math.stackexchange.com, so I voted to close.

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