I'm looking for a reference (with proof!) for the following result:
Let $A$ and $B$ be unital, associative rings.
(Eilenberg-Watt) Let $F: {}_{A}\mathrm{Mod} \to {}_{B}\mathrm{Mod}$ be a functor that is
additive and preserves direct sums and quotients. Then there is a
natural isomorphism $$\alpha: P \otimes_A \bullet \to F$$ where
$P=F(A)$ has the natural right $A$-module structure.
In short, an additive functor that preserves direct sums and quotients coincides with the functor $P \otimes_A \bullet$.
Please check if all assumptions match, because I have seen plenty of variations of this theorem but not this exact one.
Best Answer
See this question and answer for a proof. The idea is exactly the one described in the second part of Andrea's answer: for any $A$-module, you have a presentation $A^{(J)}\to A^{(I)}\to M\to 0$, which you can then use together with your assumptions to deduce that $F(M)\cong F(A)\otimes_A M$.
Note that $F(A)$ has a $B$-module structure by definition, and also a right $A$-module structure given by $F(r_a), a\in A$ where $r_a : A\to A$ is given by $x\mapsto xa$ (this is a morphism of left $A$-modules).
To get naturality of the isomorphism $F(M)\cong F(A)\otimes_A M$, the trick is to define, for any $F$ and any $M$ (no assumptions so far) a natural morphism $F(A)\otimes_A M\to F(M)$. One then uses the assumptions to prove that this natural morphism is an isomorphism.
This natural morphism is defined by the following composite: $M\to \hom(A,M) \to \hom(F(A),F(M))$ and then the universal property of the tensor product. There is one thing to note here : $\hom(A,M)\to \hom(F(A),F(M))$ is a morphism of $A$-modules because of the definition of the right $A$-module structure on $F(A)$.
This gives you a natural morphism, which is obviously an isomorphism when $M=A$, and since both sides preserve direct sums and cokernels, it is an isomorphism for any $M$.
Let me mention two additional things with respect to some points in Andrea's answer :
1- In the example he gives in the first part of the question, there is a mistake : we never consider $P\otimes_\mathbb C X$, but $P\otimes_A X$ - in particular the dimension is not the product of the dimensions.
2- There is no difference between preserving cokernels of general maps and preserving cokernels of injections: indeed if you preserve the latter, then you preserve epimorphisms, and then for a general cokernel sequence $M\to N\to C\to 0$, you can separate it as $M\to \mathrm{im}(M\to N)\to 0$ and $0\to \mathrm{im}(M\to N)\to N\to C\to 0$. This gives two a priori different definitions of right exact functors which end up agreeing.