Over a ring $R$ we have a notion of finitely presented module, namely:

Definition 1 A module $F$ is finitely presented if there are $m$, $n$ positive integers such that $R^m\to R^n\to F\to 0$ is exact.

However there is also the (more general)

Definition 2 If $\mathcal{C}$ is a category with arbitrary direct limit, an object $F$ is finitely presentable if $\mathbf{Hom}_\mathcal{C}(F,-)$ commutes with direct limits.

I have read from multiple sources that these definitions are equivalent. I was able to find a proof of $1\Rightarrow 2$ in Commutative Coherent Rings Theorem 2.1.5(2). Can somebody point to a reference or at least the idea of the proof?


closed as off-topic by Jeremy Rickard, YCor, András Bátkai, Stefan Kohl, Alex Degtyarev Jun 8 '15 at 19:10

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