In the context of Riemann Surfaces, or $\mathbb C$ manifolds (or even more generally) I wonder if a module over a c-soft sheaf $\mathcal S$ is itself c-soft which I believe implies soft if the space is Hausdorff. I tried to to show it but I'm not sure how to use the softness of $\mathcal S$.
So take $\mathcal F$ a $\mathcal S$-module and $K\subseteq X$ compact. Consider $i\colon K\hookrightarrow X$ the inclusion and $s\in \Gamma(K,i^{-1}\mathcal F)$ a global section of $i_*i^{-1}\mathcal F$. Now we have that $\Gamma(K,i^{-1}\mathcal F)=\operatorname{colim}_{U\supseteq K}\Gamma(U,\mathcal F)$ so $s$ is represented by some $t\in\Gamma(U,\mathcal F)$ which is a $\Gamma(U,\mathcal S)$-module. At this point how can we use our softness assumption of $\mathcal S$ ?