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