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

Definition 1A 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 2If $\mathcal{C}$ 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?

Locally presentable and accessible categories(esp. Example 1.2(5) and Cor 3.13) $\endgroup$ – YCor Jun 8 '15 at 17:26