TITLE: Reference request: compact objects in R-Mod are precisely the finitely-presented modules?
QUESTION [28 upvotes]: Let $R$ be a ring. According to this MO question, the modules $M \in R\text{-Mod}$ such that $\text{Hom}(M, -)$ preserves all filtered colimits (the compact objects) are precisely the finitely-presented modules. Where can I find a proof of this?

REPLY [9 votes]: This is a complement to Matt E's nice answer. Let us prove 
Lemma A. A direct summand of a finitely presented $R$-module is finitely presented.
Consider the following condition $(*)$ on a finitely generated $R$-module $C$:
$(*)$ If 
$$
0\to A\to B\to C\to0
$$
is an exact sequence of $R$-modules, and if $B$ is finitely generated, then so is $A$.
Lemma B. Condition $(*)$ holds if and only if $C$ is finitely presented.
The implication "$(*)\implies C$ finitely presented'' is clear. It is also clear that a direct summand of an $R$-module satisfying $(*)$ satisfies $(*)$. So, we are left with proving that finitely presented $R$-modules satisfy $(*)$.
Let
$$
0\to A\to B\to C\to0,\quad F\to G\to C\to0
$$
be respectively an exact sequence and a finite presentation of $C$. There are morphisms making the diagram
$$
\begin{matrix}
F&\to&G&\to&C&\to&0\\ 
\downarrow&&\downarrow&&||&&||\\
A&\to&B&\to&C&\to&0
\end{matrix}
$$
commute. By the Snake Lemma, the vertical arrows have isomorphic cokernels. QED
Now let us rephrase Matt E's proof that finitely presented $\implies$ compact. (I'll use the setup of Categories and Sheaves by Kashiwara and Schapira.)
Let $M$ be a finitely presented $R$-module and $F:I\to\mathcal M$ a functor from a small filtrant category to the category of $R$-modules (category which depends on the underlying universe). We must show that the natural map
$$
\lim_{\longrightarrow}\text{Hom}(M,F)\to\text{Hom}\left(M,\lim_{\longrightarrow}F\right)
$$
is bijective.
As $M$ is finitely presented, there is a finite category $J$ and a functor $G:J\to\mathcal M$ such that $G(j)$ is free and finitely generated for all $j$, and
$$
\lim_{\longrightarrow}G\simeq M.
$$
Then there are natural isomorphisms
$$
\lim_{\longrightarrow}\text{Hom}(M,F)
$$
$$
\simeq\lim_{\substack{\longrightarrow\\ i}}\text{Hom}\left(\lim_{\substack{\longrightarrow\\ j}}G(j),F(i)\right)
$$
$$
\simeq\lim_{\substack{\longrightarrow\\ i}}\lim_{\substack{\longleftarrow\\ j}}\text{Hom}(G(j),F(i))
$$
$$
\simeq\lim_{\substack{\longleftarrow\\ j}}\lim_{\substack{\longrightarrow\\ i}}\text{Hom}(G(j),F(i))
$$
$$
\simeq\lim_{\substack{\longleftarrow\\ j}}\text{Hom}\left(G(j),\lim_{\substack{\longrightarrow\\ i}}F(i)\right)
$$
$$
\simeq \text{Hom}\left(\lim_{\substack{\longrightarrow\\ j}}G(j),\lim_{\substack{\longrightarrow\\ i}}F(i)\right)
$$
$$
\simeq \text{Hom}\left(M,\lim_{\longrightarrow}F\right).
$$
The third isomorphism follows from the fact that, in the category of sets, small filtrant limits commute with finite projective limits. The fourth isomorphism results from the assumption that $G(j)$ is free and finitely generated. The other isomorphisms are obvious.