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?
