Kashiwara and Schapira mentioned that c-soft is a local property on their book. (The topological space is assumed to be locally compact.) That is, $\mathscr{F}$ is a c-soft if and only if there is an open cover such that $\mathscr{F}$ restricting to each of the open set in the cover is c-soft.
I'm trying to prove this fact but I feel that the only c-soft sheaves I know come from partition of unity which is something I cannot use. So I'm wondering how a sheaf can be c-soft without being fine?
Thank you.