Exercise 1.5.7 in Weibel's book about mapping cone and mapping cylinder Given a short exact sequence of chain complexes
$$0\rightarrow B \xrightarrow{\ f\ }C \xrightarrow{\ g\ }D\rightarrow 0$$
The problem asks to show that there is a quasi-isomorphism $B[-1]\rightarrow \mathrm{cone}(g)$. And showing that the composite 
$$H_{n}(D)\xrightarrow{\ \partial\ }H_{n-1}(B) \xrightarrow{\ \simeq\ }H_{n}(\mathrm{cone}(g))$$
is the usual map induced by the inclusion of $D$ in $\mathrm{cone}(g)$.
The only map I can image from $B[-1]$ to $\mathrm{cone}(g)$ is given by $b\mapsto (f(b), 0)$. However, I can't show that it is a quasi-isomorphism directly or put it into a commutative digram of short exact sequences. Can anyone give me some hints?
I am pretty sure that we will use the short exact sequence
$$0\rightarrow C \rightarrow \mathrm{cone}(f) \rightarrow B[-1] \rightarrow 0$$
other two short exact sequences that might be helpful are
$$0\rightarrow D \rightarrow \mathrm{cone}(g) \rightarrow C[-1] \rightarrow 0$$
and 
$$0\rightarrow C \rightarrow \mathrm{cyl}(g) \rightarrow \mathrm{cone}(g)  \rightarrow 0$$
Thank you very much.
 A: A quasi-isomorphism $\psi:B[-1]\to\mathrm{cone}(g)$ is $b\mapsto(-f(b),0)$.
We do indeed use some of your suggested short exact sequences. Consider the diagram
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!}
\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.}
%
\begin{array}{ccccccccccccc}
&&&&0&\ra{}&B[-1]&\ra{f}&C[-1]&\ra{g}&D[-1]&\ra{}&0\\
&&&&&&\da{\psi}&&\da{\mathrm{id}}&&&&\\
&&0&\ra{}&D&\ra{\iota}&\mathrm{cone}(g)&\ra{\delta}&C[-1]&\ra{}&0\\
&&&&\da{}&&\da{\mathrm{id}}&&&&&&\\
0&\ra{}&C&\ra{}&\mathrm{cyl}(g)&\ra{}&\mathrm{cone}(g)&\ra{}&0
\end{array}
$$
which creates homology long exact sequences
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!}
\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.}
%
\begin{array}{ccccccccccccc}
\cdots&\ra{}&H_{n+1}(C[-1])&\ra{}&H_{n+1}(D[-1])&\ra{\partial}&H_{n}(B[-1])&\ra{}&H_{n}(C[-1])&\ra{}&\cdots\\
&&\da{\mathrm{id}}&&\da{\mathrm{id}}&&\da{\psi_{*}}&&\da{\mathrm{id}}&&\\
\cdots&\ra{}&H_{n+1}(C[-1])&\ra{\partial}&H_{n}(D)&\ra{\iota_{*}}&H_{n}(\mathrm{cone}(g))&\ra{}&H_{n}(C[-1])&\ra{\partial}&\cdots\\
&&\da{\mathrm{id}}&&\da{\cong}&&\da{\mathrm{id}}&&\da{\mathrm{id}}&&\\
\cdots&\ra{\partial}&H_{n}(C)&\ra{}&H_{n}(\mathrm{cyl}(g))&\ra{}&H_{n}(\mathrm{cone}(g))&\ra{\partial}&H_{n-1}(C)&\ra{}&\cdots
\end{array}
$$
By the five lemma, $\psi$ is a quasi-isomorphism. By the commutativity of the homology diagram at the top middle square,
$$
\partial:H_{n}(D)\to H_{n-1}(B)
$$
is the map
$$
H_{n}(D)\xrightarrow{\iota_{*}}H_{n}(\mathrm{cone}(g))\cong H_{n-1}(B),
$$
where $\iota_{*}$ is the map induced by the inclusion $D\xrightarrow{\iota}\mathrm{cone}(g)$.
