I'm learning about the following facts:
$1)$ If a functor $\mathcal{F}$ is right adjoint to $\mathcal{G}$, then $\mathcal{F}$ preserves limits and $\mathcal{G}$ preserves colimits. In particular for abelian categories, $\mathcal{F}$ preserves kernels and $\mathcal{G}$ preserves cokernels.
$2)$ For a commutative ring with unit $A$ and $M$ an $A$-module, then $\mathcal{H}:=\hom_A(M,-)$ is right adjoint to $\mathcal{T}:=\,-\otimes_A M$. In particular, $\mathcal{H}$ preserves kernels and $\mathcal{T}$ preserves cokernels.
$3)$ $\hom_A(M, -)$ is exact $\implies$ $-\otimes_A M$ is exact (or: $M$ is projective $\Rightarrow M$ is flat)
Is it possible to prove $3)$ directly from $1)$ and $2)$?
Or more generally: if I have and adjoint pair $(\mathcal{F}$, $\mathcal{G})$ with one of them exact, can I conclude something about the exactnesse of the other?
Very interesting question! I don't think that one gets anything in general, unfortunately. One important class of examples is sheafification functors. The left adjoint to the inclusion of any category of sheaves into presheaves preserves finite limits (which is often referred to as being left exact, even in the non-additive case.) However, its right adjoint, the inclusion functor itself, rarely preserves finite colimits. This is the familiar result, in the additive case, that the cokernel of a map of sheaves is not generally a sheaf.