Is this Sequences of Complexes of Sheaves Exact? So in  another question of mine there is a sequence of complexes of sheaves which the author asserts is exact.  
Let $K^{\bullet} = \underline{\mathbb{C}}^* \  \underrightarrow{d\  log} \  \underline{A}^1_{M, \mathbb{C}}$  and so we have an exact sequence of complexes of sheaves:
$$0 \rightarrow {\mathbb{C}}^* \rightarrow K^{\bullet}
 \rightarrow \underline{A^2}{M, cl}[-1] \rightarrow 0 $$
Where that nastily noted $\underline{A^2}{M, cl}[-1]$ means the two term complex with 0 in the first slot and closed 2 forms on $M$ in the second slot.
The fact that this sequence is exact in itself seems to rely on the fact that the sheafification of the image of the contant $\mathbb{C}^*$ sheaf is isomorphic to the sheaf of smooth functions $\underline{\mathbb{C}^ * }$ right?  Well that part is bothersome to me
 A: This is not related to sheafification. The sheaf $\mathbb{C}^{*}$  of locally constant functions on $M$ is already a sheaf,  so sheafification will not change it. 
This  sequence is not an exact sequence of complexes but it is an exact triangle of complexes. That is - it is an exact sequence of complexes, up to quasi-isomorphism.  The obvious short  exact sequence of complexes is 
$$
0 \to \mathbb{C}^{*} \to \left[\begin{array}{c} \underline{\mathbb{C}}^{*} \\ \downarrow \\ A^{1}_{M} \end{array}
\right] \to \left[ \begin{array}{c} A^{1}_{M,cl} \\ \downarrow \\ A^{1}_{M} \end{array}\right] \to 0 .
$$
Now note that the last complex has an obvious surjective map 
$$
\left[ \begin{array}{c} A^{1}_{M,cl} \\ \downarrow \\ A^{1}_{M} \end{array}\right] \to \left[ \begin{array}{c} 0 \\ \downarrow \\ A^{2}_{M,cl} \end{array} \right]
$$
and that this surjective map is a quasi-isomorphism. So up to a quasi-isomorphism, you can replace the last term in the short exact sequence with the complex you wanted.
