Let $\textbf P$ be projective resolution of a module. Why is the sequence $P_{n+1}\xrightarrow{\partial_{n+1}}P_n\xrightarrow{\overline{\partial}}B_{n-1}\rightarrow 0$ exact, where $B_{n-1}=B_{n-1}(\textbf P)$ and $\partial_n$ factors as $P_n\xrightarrow{\overline{\partial}} B_{n-1}\xrightarrow{i} P_{n-1}$?
I dont't see why $\ker \overline{\partial}\subseteq \mathrm{im}\ \partial_{n+1}$.