アーベル圏にほけるホモロジー

まず$kerg,cokerf,imf$をとる.
\begin{xy} \xymatrix { Kerg \ar[dr]^{kerg} && Cokerf \\ & B \ar[ur]^{cokerf} \ar[dr]^g \\ Imf \ar[ur]^{imf} & A \ar[r]_0 \ar[u]^f \ar[l]_q & C } \end{xy}

$g\circ f=f\circ imf \circ q=0$と$q$がepiより,$g\circ imf=0$なので,$(Kerg,kerg)$の普遍性より,
$\exists! \bar{f}:Imf\rightarrow Kerg$
$s.t.\quad imf=kerf\circ\bar{f}$
$g\circ f=0$なので,
$(Cokerf,cokerf)$の普遍性より,
$\exists! \bar{g}:Cokerf\rightarrow C$
$s.t. \quad g=\bar{g}\circ cokerf$
\begin{xy} \xymatrix { Kerg \ar[dr]^{kerg} && Cokerf\ar@[green]@{.>}[dd]^{\textcolor{green}{\bar{g}}} \\ & B \ar[ur]^{cokerf} \ar[dr]^g \\ Imf \ar[ur]^{imf} \ar@[green]@{.>}[uu]^{\textcolor{green}{\bar{f}}} & A \ar[r]_0 \ar[u]^f \ar[l]_q & C } \end{xy}
$h\coloneqq cokerf\circ kerg$とおく.
$h\circ\bar{f}\circ q=cokerf\circ kerg\circ\bar{f}\circ q$
$\qquad\quad\ \;=cokerf\circ imf\circ q$
$\qquad\quad\ \;=cokerf\circ f$
$\qquad\quad\ \;=0$
で,$q$はepiより$h\circ\bar{f}=0$なので,
$(Coker\bar{f},coker\bar{f})$の普遍性より,
$\exists!\bar{h}:Coker\bar{f}\rightarrow Cokerf$
$s.t.\quad h=\bar{h}\circ coker\bar{f}$

$\bar{g}\circ\bar{h}\circ coker\bar{f}=\bar{g}\circ h$
$\qquad\qquad\qquad\,=\bar{g}\circ cokerf\circ kerg$
$\qquad\qquad\qquad\,=g\circ kerg$
$\qquad\qquad\qquad\,=0$
で,$coker\bar{f}$はepiより,$\bar{g}\circ \bar{h}=0$なので,
$(Ker\bar{g},ker\bar{g})$の普遍性より,
$\exists!\widetilde{h}:Coker\bar{f}\rightarrow Ker\bar{g}$
$s.t.\quad \bar{h}=ker\bar{g}\circ \widetilde{h}$
\begin{xy} \xymatrix {Coker\bar{f} \ar@[red]@{.>}[rrd]^{\textcolor{red}{\bar{h}}} \ar@[red]@{.>}[rr]^{\textcolor{red}{\widetilde{h}}} &&Ker\bar{g} \ar[d]_{ker\bar{g}} \\Kerg \ar[dr]^{kerg} \ar@[red][rr]^{\textcolor{red}{h}} \ar[u]^{coker\bar{f}} && Cokerf \ar@[green]@{.>}[dd]^{\textcolor{green}{\bar{g}}} \\ & B \ar[ur]^{cokerf} \ar[dr]^g \\ Imf \ar[ur]^{imf} \ar@[green]@{.>}[uu]^{\textcolor{green}{\bar{f}}} & A \ar[r]_0 \ar[u]^f \ar[l]_q & C } \end{xy}
\begin{xy} \xymatrix { Imf \ar@[green]@{.>}[r]^{\textcolor{green}{\bar{f}}} \ar[d]_{id_{Imf}}^{\textcolor{blue}{epi}} &Kerg \ar[r]_{coker\bar{f}} \ar[d]_{kerg}^{\textcolor{blue}{mono}} &Coker\bar{f} \ar[r]^0 \ar@[red]@{.>}[d]_{\textcolor{red}{\bar{h}}} &0 \ar[d]_{id_0}^{\textcolor{blue}{mono}} \\ Imf \ar[r]_{imf} &B \ar[r]_{cokerf} &Cokerf \ar[r]_0 &0 } \end{xy}
なので,4項補題より,$\bar{h}$はmono.
\begin{xy} \xymatrix { 0 \ar[r]^0 \ar[d]_{id_{0}}^{\textcolor{blue}{epi}} &Kerg \ar[r]^{kerg} \ar[d]|{\textcolor{red}{\widetilde{h}}\circ coker\bar{f}} &B \ar[r]^g \ar[d]_{cokerf}^{\textcolor{blue}{epi}} &C \ar[d]_{id_C}^{\textcolor{blue}{mono}} \\ 0 \ar[r]_{0} &Ker\bar{g} \ar[r]_{ker\bar{g}} &Cokerf \ar[r]_{\bar{g}} &C } \end{xy}
なので,4項補題より,$\widetilde{h}\circ coker\bar{f}$はepi.
よって$\widetilde{h}$はmonoかつepiなので,同型射.