The perfection of a ring $A$ of prime characteristic $p$ is the perfect ring $A_\rm{pf}=$ lim{$A\to A\to ...$} where all maps are Frobenius. It does not commute with products, as was shown by YCor in the answer to this question. So, as it was suggested in that answer, I ask as a separate question the following one:

Why in a scheme $X$ (not necessarily reduced), the presheaf associated to $U\to \Gamma (U,\mathcal{O_\rm{X}})_\rm{pf}$ is a sheaf?

This is stated in section 6 in Greenberg: Perfect closures of rings and schemes but since perfection does not commute with products, I do not understand the proof.