I've cleaned up the comment thread of Are there two non-isomorphic modules such that all the Hom-sets are isomorphic?. Here's a snapshot:

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This follows from Yoneda lemma if you assume this isomorphisms are natural. – Piotr Pstrągowski 21 hours ago

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Among the worst titles in my memory. – Igor Rivin 20 hours ago

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And if you do not assume the isomorphisms between Hom-sets to be natural, then for example over a field the question boils down to whether it is possible for two non-isomorphic vector spaces to have isomorphic duals. Over the field with two elements this is simply a question about the cardinality of power sets, which might very well be independant of ZFC. – Piotr Pstrągowski 20 hours ago

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@Igor: I guess it would be nice to edit the title and then inform the OP, so that next time he poses a question he would be more precise. – Chandrasekhar 12 hours ago

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Flagged Igor's comment as offensive and hate speech. – Guntram 11 hours ago

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@Guntram: You keep using that word. I do not think it means what you think it means. – Harry Gindi 9 hours ago

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I suggest "Are modules isomorphic if their Hom-sets are all isomorphic?" (or something like that). – Mark Grant 5 hours ago

@Piotr: could you please explain the "boils down" a bit further? The implication "isomorphic duals" ⟹ "all hom spaces isomorphic" seems to require some implication of the sort 2κ=2λ⟹ακ=αλ for all cardinals α. Is this true? – a-fortiori 5 hours ago

Actually, the original title was "If the homomorphisms are isomorphic then they are isomorphic"