For given $A$-modules and homomorphisms $M'\stackrel{u}\to M\stackrel{v}\to M''\to 0$ this is an exact sequence iff for all $A$-modules $N$, the sequence $$0\to\operatorname{Hom}(M'',N)\stackrel{\overline{v}}\to \operatorname{Hom}(M,N)\stackrel{\overline{u}}\to \operatorname{Hom}(M',N)$$ is exact.
How to prove exactness at $\operatorname{Hom}(M,N)$ ?
Any help would be appreciated.