An object $C$ in an additive category admitting all filtered direct limits $\mathcal{C}$ is called "of finite type" if the canonical map $$\underrightarrow{\lim} Hom_{\mathcal{C}}(C,F(i))\to Hom_{\mathcal{C}}(C,\underrightarrow{\lim}F)$$ is injective for every $I$ directed poset for every functor $F:I\to \mathcal{C}$
In the case $\mathcal{C}$=Mod-R prove that this definition is equivalent to the definition of "finitely generated"
The exercise has this strange hint: Use the fact that if $\mathcal{F}$ is the set of finitely generated submodules of a module C then $$C/{\sum_{A\in\mathcal{F}}A}=\underrightarrow{\lim}C/A$$
I say that the hint is strange because $\displaystyle {\sum_{A\in\mathcal{F}}A}=C$
You're right in that $\sum_A A = C$, so $C/\sum_A A={\lim\limits_{\rightarrow}}_A C/A=0$, and that's exactly the colimit you need to instantiate the given condition with. The right hand side being zero, any element of $\varphi\in\text{Hom}(C,C/A)$ must eventually vanish in some $\text{Hom}(C,C/B)$ for $A\subseteq B$. What's a natural choice for $A$ and $\varphi$ to consider here?