Let $ P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \xrightarrow{d_0} M \to 0$
be an exact sequence of $R$-modules. Consider
$ (*) \hspace{1 cm} P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \to 0$
that is, the sequence with $M$ removed. Then this resulting sequence should still be exact at $P_1$ since we did not change the maps $d_2, d_1$. Now apply the left exact functor $\mathrm{Hom}(-, N)$ (for some $R$-module $N$) to get
$ (**) \hspace{1 cm} 0 \to \mathrm{Hom}(P_0, N) \xrightarrow{\overline{d_1}} \mathrm{Hom}(P_1, N) \xrightarrow{\overline{d_2}} \mathrm{Hom}(P_2, N) $
Clearly, $\overline{d_1}$ does not have to be injective since $(*)$ was not exact at $P_0$. What I'm not so clear about is why, even though $(*)$ was exact at $P_1$, we also don't necessarily get exactness at $\mathrm{Hom}(P_1, N)$ anymore.
Can you give me a simple example with concrete $R$-modules $P_1, M, N$ such that $(*)$ is exact at $P_1$ but $(**)$ not exact at $\mathrm{Hom}(P_1, N)$? Thanks.