TITLE: Exactness of an additive left Kan extension
QUESTION [6 upvotes]: Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily described if we see  $\phi$ as an additive functor between the one-object categories $R$ and $S$ and we view a left $S$-module $M$ as an additive functor $M:S\to Ab$. In this case, $\phi^*$ is just composition by $\phi$, $(M:S\to Ab)\mapsto (M\circ\phi:R\to Ab)$.
By definition of flat homomorphism, $\phi_!$ is an exact functor, while $\phi^*$ is exact for any ring homomorphism $\phi$.
Consider now the categories of finitely presented modules $fp(R)$ and $fp(S)$, and restrict the functor $\phi_!$ to an additive functor $F:fp(R)\to fp(S)$. Denote also by $fp(R)-Mod$ and $fp(S)-Mod$ the categories of additive functors $fp(R)\to Ab$ and $fp(S)\to Ab$, respectively. Of course, also here there is an induced exact functor 
$$F^*:fp(S)-Mod\to fp(R)-Mod$$
and, using additive Kan extensions one may construct a left adjoint 
$$F_!:fp(R)-Mod\to fp(S)-Mod.$$
Can we prove that the functor $F_!$ is exact?
Spelling this out, rember that, given a sequence of left $fp(R)$-modules
$$0\to M_1\to M_2\to M_3\to 0$$
this sequence is exact if and only if, for all $P\in fp(R)$, the sequence of abelian groups
$$0\to M_1(P)\to M_2(P)\to M_3(P)\to 0$$
is short exact. Thus one should prove (or disprove) that for any short exact sequence as above, 
$$0\to F_!M_1(Q)\to F_!M_2(Q)\to F_!M_3(Q)\to 0$$ 
is short exact for all $Q\in fp(S)$.

REPLY [5 votes]: Daniel Schäppi's answer made me realize that I actually can say something about this. I'll keep Daniel's notation.
Something more than what Daniel said is true, and it holds in a more general context: Given that $\mathcal{A}$ has finite limits, $F$ is left exact iff $\mathrm{Lan}_Y YF$ is. This is shown by Kelly (mimic Daniel's argument and use Kelly's Thm 6.11 (i) and (v) to show that a functor $\mathcal{A} \to \mathcal{V}$ is left exact iff its extension is) at the generality of categories enriched in a locally finitely presentable symmetric monoidal closed $\mathcal{V}$.
This property of the class of finite limits (that if a functor $F$ between categories with finite limits preserves all finite limits, then so does $F_!$) is a weak version of what is called soundness by Adamek, Borceux, Lack, and Rosicky in the $\mathrm{Set}$-enriched context. Other classes of limits which have this property are the class of finite products, and the class of limits indexed by categories with fewer that $\alpha$ morphisms for some fixed regular cardinal $\alpha$. This criterion was used to generalize soundness to the enriched context in Day and Lack and Lack and Rosicky. A related, stronger condition, concerning the exactness of left Kan extensions of functors $F: \mathcal{A} \to \mathcal{V}$ where $\mathcal{A}$ can be arbitrary, is studied in the enriched context by Dostal and Velebil.
One important reason to study soundness is that it allows a theory of categories which are "locally presentable with respect to a class of limits". The classical case of locally $\alpha$-presentable categories is an example (when the class of limits is those indexed by diagrams with $<\alpha$ morphisms), and another example is the theory of (many sorted) Lawvere theories: a category of models of a Lawvere theory is precisely a category which is locally presentable with respect to the class of finite products. In the enriched case especially, there are potentially other interesting examples waiting to be investigated.