Let $R$ be a commutative ring and let $L, M, N$ be three $R-$modules. Given an exact sequence of $R-$modules $$0 \rightarrow L \xrightarrow{\alpha} M \xrightarrow{\beta} N \rightarrow 0.$$ Show that if $T$ is a free $R-$module, then one obtains an exact complex $$0 \rightarrow \textrm{Hom}_{R}(T,L) \xrightarrow{\alpha \circ} \textrm{Hom}_{R}(T,M) \xrightarrow{\beta \circ} \textrm{Hom}_{R}(T,N) \rightarrow 0.$$
Edit: How to explain for every $R-$module you obtain the same result except the $\rightarrow 0$ part ? While studying for my preliminaries I came across this question. Any help in solving this is much appreciated.
Note that $\textrm{Hom}_{R}(T,-)$ is a covariant functor and left exact functor. In particular, suppose that $T$ is a free $R$-module. Then $T$ is also a projective module, and then $\textrm{Hom}_{R}(T,-)$ is a covariant functor and exact functor, since $\textrm{Hom}_{R}(T,-)$ maps surjective morphisms to surjective morphisms.