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If $R$ is a noncommutative ring, then Morita theory tells us that $R$ cannot in general be recovered from its category $\text{Mod}(R)$ of modules; that is, there can be a ring $R'$, not isomorphic to $R$, such that $\text{Mod}(R) \cong \text{Mod}(R')$. This means, for example, that “free” is not a categorical property of modules, since it depends on a choice of ring $R$, or equivalently on a choice of forgetful functor.
It’s therefore something of a surprise that “finitely presented” is a categorical property of modules, and hence that it does not depend on a choice of ring $R$. The reason is that being finitely presented is equivalent to a categorical property called compactness.