How does one dualize a short exact sequence into a long exact sequence?

Question

I can across the following in a proof I was reading but don't exactly understand how this is done/ come about. Can anyone explain?

Note: Here $M^{\ast}=Hom_{R}(M, R)$

Answer 1

Dualizing means applying the functor $\mathrm{Hom}(\bullet,R)$. This is left exact but not right exact. So while $N \to M$ is surjective $\mathrm{Hom}(N,R) \to \mathrm{Hom}(M,R)$ is not necessarily. The $\mathrm{Ext}$ functor is the right derived functor of the $\mathrm{Hom}$ functor.