I read the following in a book. If A be an abelian group and End A be defined as set of all endomorphism f :A---->A , End A along sum of functions produce an abelian group. My problem is as following.

a,b belong to A , (f+g)(ab) = f(ab)+g(ab) = f(a)f(b)+g(a)g(b)

(f+g)(a) (f+g)(b)=(f(a)+g(a))(f(b)+g(b))

Does (f+g)(ab) equal to (f+g)(a) (f+g)(b)? In other words does f+g belong to End A?