The definition of (right-/left-) exact functors is that they preserve (right-/left-) exactness of SESs. However, for some certain nice functors, as $\def\Hom{\text{Hom}\,}\Hom (A,-)$ and $A\otimes-$ we have even more: namely that if a SES sequence $$\mathcal S:0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$$ of modules remains (right-/left-) exact after homing (or tensoring) with any module A, then the sequence $\mathcal S$ was exact to begin with. The proof I have of these two examples is made in a rather ad-hoc manner and I was wondering if there is a more general proof.
To illustrate this, let me give a sketch of how we show that if $\Hom(A,\mathcal S)$ is exact at $\Hom(A,M')$, then $\mathcal S$ is exact at $M'$. Suppose not; then consider the sequence $\Hom(N,\mathcal S)$, where $N$ is the kernel of $M'\rightarrow M$. It is not exact.
Another example that I'd like to add is that of taking the stalk of a sheaf . Since this is a right-exact functor, a SES in sheaves $$0\rightarrow\mathcal F\rightarrow \mathcal G \rightarrow 0 \rightarrow 0 \tag{*}$$ would yield a right-exact sequence $$0\rightarrow\mathcal F_p\rightarrow \mathcal G_p \rightarrow 0. \tag{**}$$
Now, I'm tempted to say that if (**) is exact for all $p$ then (*) was in fact exact to start with because this is very similar to the property that I alluded to above. This is in fact true, as I tried to show in my other post. My question is: is there a purely categorical proof that generalizes all these facts?
I'd even conjecture that, under some exactness properties, if a bifunctor $\mathcal B(-,-)$ is right-/left- exact adjoint in its second argument, then the functor $\prod_A \mathcal B(A,-)$ (i.e. $\mathcal B$ parametrized by a family) reflects exactness (i.e. it enjoys the property that if $\prod_A \mathcal B(A,\mathcal S)$ is exact, then $\mathcal S$ was exact).