There are some different definitions for fine sheaves.

Let X is a paracompact Hausdorff space, a sheaf F over X is a fine sheaf, if

a) Hom(F,F) is soft

b) For every two disjoint closed subsets A,B$\subset$X, A$\cap$B=$\emptyset$ , there is an endomorphism of the sheaf F$\to$ F which restricts to the identity in a neighborhood of A and to the 0 endomorphism in a neighborhood of B.

c) For every open cover {U$_i$} of X, there is a partition of unity 1= $\sum$ f$_i$ (where the sum is locally finite) subordinate to this covering.

The definition a) appears in the Bredon's book *Sheaf theory* and you can see the definitions b) and c) at nLab here.

Can you show they are equivalent?

Here is the unanswered same question.