TITLE: Is every additive, left exact functor isomorphic to a hom functor?
QUESTION [9 upvotes]: Let $A$ be an Artin algebra, $\text{mod}\,A$ the category of finitely generated $A$-modules and $\text{Ab}$ the category of abelian groups. Is every additive, covariant, left-exact functor $F:\text{mod}\,A \rightarrow \text{Ab}$ (natural) isomorphic to $\text{Hom}(X,-)$ for some $X\in \text{mod}\,A$? How do we obtain $X$?

REPLY [6 votes]: $\DeclareMathOperator{\Set}{Set}
\DeclareMathOperator{\El}{El}
\DeclareMathOperator*{\colim}{colim}
\newcommand{\fC}{\mathcal C}$By abstract nonsense, for any category $\fC$, a functor $F:\fC\to \Set$ is a colimit of corepresented functors. Explicitly, let $\El(F)$ be the category of elements of $F$, whose objects are pairs $(c,x\in F(c))$ and morphisms $(c,x)\to (d,y)$ are morphisms $f:c\to d$ in $C$ such that $F(f)(x) = y$. There is an evident functor $p:\El(f)\to C$ which forgets the additional datum $x$, and we have $F\cong \colim_{\El(F)^{op}} y\circ p$, where $y:C^{op}\to \operatorname{Fun}(C,\Set)$ is the (opposite) Yoneda embedding.
In your situation, the functor $F$ preserves finite products (by additivity) and pullbacks (by left exactness), hence all finite limits. In this case, the index category $\El(F)$ is co-filtered, i.e. any functor $G:I\to \El(F), i\mapsto (c_i,x_i)$ extends to a cone $G_\triangleleft:I_\triangleleft\to\El(F)$: Let $c = \lim_I p\circ i$ be the limit of the underlying diagram and $x\in F(c)$ be a preimage of $\{x_i\}_{i\in I}$ under the isomorphism $F(c)\cong \lim_I F\circ G\subset \prod_{i\in I} F(c_i)$, so that sending the cone point to $(c,x)$ and the cone legs to the lifts of the projections from the limit $c\to c_i$ is the required extension. (Note that if $\mathcal C$ has all products and $F$ preserves them, you can take an arbitrary small category for $I$. If you could take $\El(F)$ itself, this means that $\El(F)$ has an initial object $(c_0,x_0)$, and unraveling the definitions shows that $F$ is corepresented by $c_0$. The adjoint functor theorem in Dan Petersen's answer essentially works this way.)
On the other hand, since colimits in functor categories are computed objectwise and filtered colimits commute with finite limits in Sets, any filtered colimit of corepresented functors preserves finite limits. So abstractly, the best thing you can hope for is a pro-object representing your functor, i.e. a diagram $C:I\to \fC$ from a co-filtered category such that $F(c') \cong \colim_{I^{op}} \fC(C(i),c')$. Here you can always take $I$ to be the opposite of a cardinal, considered as a poset, by choosing a cofinal subset of $\El(F)$. For example, the functor $V\mapsto V^{\oplus\infty}$ is corepresented by the pro-object with $I = \mathbb N^{op}$ and $C(n) = k^n$, with the maps $C(n+1)\to C^n$ given by projection. On the other hand, in Zhen Lin's example the category of elements $\El(F)$ has cofinality $\kappa$, which is bigger than the cofinality of any slice category, so that the functor can not be representable.
Restrict now to the setting in the question, i.e. $\mathcal C$ is the category of finitely-generated modules over an Artin algebra $R$.
We know that $F = \colim_{I^{op}} y(M_i)$ is a subfunctor of a corepresentable functor $y(M)$. This means that there are compatible maps $f_i:M\to M_i$. The submodules $\ker f_i$ of $M$ form a descending chain; since $R$ is Artinian and $M$ finitely generated, they stabilize at a finite level $i$. Replacing $M$ by $M/\ker f_i$ and $I$ by $I_{i/}$, we may assume that all the maps $f_i$ are injective. Similarly, the images of the maps $M_{i'}\to M_i$ form a descending chain of submodules of $M_i$, and therefore stabilize at a finite stage. Let $\{M'_i\}_{i\in I}$ be the pro-object of these images, and $F' = \colim_{I^{op}} y(M'_i)$, so that we get a factorization $F\Rightarrow F'\Rightarrow y(M)$. The first map is an isomorphism (injectivity and surjectivity can be checked by lifting elements a finite amount through the colimit). Replacing $\{M_i\}_{i\in I}$ by $\{M'_i\}_{i\in I}$, we may assume that all transition maps are surjective. Thus all transition maps in the colimit $F'(M') = \colim_{I^{op}} \fC(M_i,M')$ are injective, so that $y(M_i)$ is a subfunctor of $F$ and hence of $y(M)$, which is equivalent to the maps $M\to M_i$ being surjective. Together with injectivity, this shows that we may assume that $\{M_i\}_{i\in I} = \{M\}_{i\in I}$ is a constant pro-object, which is equivalent to corepresentability of $F$.