Pure monomorphism of functors-

Question

Suppose that $F, G: Q\rightarrow R{\rm -Mod}$ be two covariant functors where $Q$ is an abelain category and $R$ is a commuatative ring. Also let $\eta: F\rightarrow G$ be a natural transformation such that $\eta_v: F(v)\rightarrow G(v)$ is a pure monomorphism for every object $v$ of $Q$. Hence $\eta_v^+:G(v)^+\rightarrow F(v)^+$ is an split epimorphism for every object $v$ where by $(-)^+$ we mean ${\rm Hom}(- , \frac{Q}{Z})$.

One can show that $F^+, G^+: Q\rightarrow R{\rm -Mod}$ are both functors.

Question: There is a natural transformation $\eta^+: G^+\rightarrow F^+$ such that for each object $v$ of $Q$ we have a split epimorphism $G^+(v)\rightarrow F^+(v)$ of modules. Can we deduce that $\eta^+$ is split? i.e. Is there any natural transformation $\beta: F^+(v)\rightarrow G^+(v)$ s.t. $\eta^+ \circ \beta = 1_{F^+}$?

Answer 1

If you take $R$ to be, say, a finite field, so that all epimorphisms and monomorphisms of $R$-modules are pure and split, and + is just vector space duality, and for simplicity restrict to functors that take values in finite-dimensional vector spaces, then your question reduces to asking whether a monomorphism (or epimorphism) of such functors must be split.

This is certainly not true in general. For example, let $A$ be a finite-dimensional $R$-algebra, and $M\to N$ a non-split epimorphism of $A$-modules. Then $\operatorname{Hom}_A(N,-)\to\operatorname{Hom}_A(M,-)$ is a non-split monomorphism of functors from finitely generated $A$-modules to finite-dimensional $R$-modules.