on page $38$ of the book "Ring theory" by Louis Rowen it is shown that $R-\mathbf{Mod}$ and $M_{n}(R)-\mathbf{Mod}$ are equivalent categories ($R$ a ring). Some lines below the author states that these categories are not isomorphic, but why is this?
Equivalent categories yet not isomorphic
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category-theory
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6I don't think that is correct. (Modulo set theory, I guess...) the only obstruction to two equivalent categories to being isomorphic is that they have different number objects in each isomorphism class, but categories of modules are too large for this to happen; with a sufficiently good set-theoretic foundation —universes, or something— one should be able to construct an iso from every equivalence. What *is* true is that none of the equivalences we usually deal with are isomorphisms. – 2012-02-28
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0Should it be $R$-Mod and $M_n(R)$-Mod? – 2012-02-28
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0@Arturo Magidin: Oh, sorry! yes $M_{n}(R)-Mod$. – 2012-02-28
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0@Mariano: We'd need a sufficiently large axiom of choice, but then again, we need that to prove that full + faithful + essentially surjective on objects is enough to be (part of) an equivalence of categories. – 2012-02-28
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0@ZhenLin, ah, indeed! – 2012-02-28
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0What is Mn(R)-Mod? I mean: what is the full name? – 2012-02-29
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0@magma: It is most likely the category of left modules over the ring of $n\times n$ matrices with entries in $R$. – 2012-10-19