For $A$-modules and homomorphisms $0\to M′\stackrel{u}{\to}M\stackrel{v}{\to}M′′\to 0$ is exact iff for all A-modules N, the sequence
$0 \to Hom(M′′,N)\stackrel{\bar{v}}{\to} Hom(M,N)\stackrel{\bar{u}}{\to} Hom(M′,N)$ is exact.
How $\bar{u}$ and $\bar{v}$ are defined ? How do I prove that the sequence is exact at Hom(M′′,N) and Hom(M′,N)?
thanks for any help or hints
Writing composition from left to right, $\bar v$ maps $\phi\mapsto v\cdot \phi$, similarly for $\bar u$. For exactness, you need to show that $\ker\bar u={\rm im\,}\bar v$ and that $\bar v$ is injective. For example, the injectivity follows because $v$ is epimorphism, so $v\cdot\phi= 0$ implies $\phi=0$.