I have a question regarding the proof of Cartan's theorem that I found in the following document (last page). I just recently started self-studying cohomology so please bear with me.
First, when he says "we now take the long exact sequence in cohomology associated to the short exact sequence of complexes $0\rightarrow \check{C}^{\cdot}(X,\mathcal{F})\rightarrow \check{C}^{\cdot}(X,\mathcal{G})\rightarrow \check{C}^{\cdot}(X,\mathcal{H})\rightarrow 0$", how is this long exact sequence constructed ? My guess is that we'll use the snake lemma somehow but could anyone explain how that's exactly done ?
My second question is about the vertical arrows in the last diagram:
$\require{AMScd}$
$\begin{CD}
...@>>>\check{H}^{i-1}(X,\mathcal{G})@>>>\check{H}^{i-1}(X,\mathcal{H})@>>>\check{H}^{i}(X,\mathcal{F})@>>>\check{H}^{i}(X,\mathcal{G})@>>> ...\\
\ @VVV @VVV @VVV @VVV\\
...@>>>H^{i-1}(X,\mathcal{G})@>>>H^{i-1}(X,\mathcal{H})@>>>H^{i}(X,\mathcal{F})@>>>H^{i}(X,\mathcal{G})@>>> ...
\end{CD}$
How are these vertical arrows defined ?
Thank you in advance for your patience and your answers!