Infinite direct sum of sheaf of modules is same as sheaf formed by infinite direct sum of modules?

This is a statement made in Mumford, Oda Algebraic Geometry II.(Chpt 1, Sec 2. Prop 2.7) http://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf

If $$M_a$$ is any collection of $$R$$ modules and denote $$X=Spec(R)$$, then $$\overline{\oplus_aM_a}=\oplus_a\overline{M_a}$$ where $$\overline{M}$$ denotes formation of sheaf of $$O_X$$ modules.

The proof goes along by using $$Spec(R_f)$$ affine which enforces quasi-compact. Then both sides of equation are canonically identified.(It is not hard to check this equality.) Then extension to a unique isomorphic sheaf by $$\mathcal{B}$$-sheaf extension.

$$\textbf{Q:}$$ Isn't this is sort of saying infinite direct sum of quasi-coherent sheaves of modules is quasi-coherent over affine scheme? Or am I misunderstanding the statement.

$$\textbf{Q':}$$ What is the example of infinite direct sum of sheaves of free module over affine scheme fails to be a sheaf?