Let $Sh_{et}(Spec(A),\mathbb{F}_p)$ be the category of $p$-torsion affine etale sheaf on $Spec(A)$, which is a contravariant functor from AFFINE etale covers of $Spec(A)$ to $\mathbb{F}_p$-modules satisfying the axioms of sheaves and assume $f:A\to B$ is an etale morphism of rings. I want to show that the pullback functor $f^*:Sh_{et}(Spec(A),\mathbb{F}_p)\to Sh_{et}(Spec(B),\mathbb{F}_p)$ satisfies the "solution set criterion", which concretly means that for a functor $G:C\to C'$ and any object $x$ in $C'$, there exists an index set $I$, a family of objects $\{a_i|i\in I\}$ in $C$ and arrows $f_i:x\to G(a_i)$ such that for every $x\to G(a)$ where $a$ is in $C$, it can be factored as $G(t)f_i$ for some $i\in I$ and some arrow $t:a_i\to a$.
I want $I$ to be the singleton and $a_i=a_0=f_*x$. There is a morphism $f^*f_*x\to x$ and hence $f^*f_*x\to f^*a$. This is equivalent to $f_*x\to f_*f^*a$. But I cannot continue because even if $f$ is etale I cannot find $f_*f^*a\to a$. Can anyone give me other ideas?