= Solution
On every open set $U$, define $F_U^\#(a)=a^p$. In <characteristic of a ring> $p$, the <binomial theorem> gives $(a+b)^p=a^p+b^p$, so these are <ring homomorphisms>; they commute with restrictions and hence define a morphism of <sheaves of rings>.
On an affine chart $\operatorname{Spec}A$, the inverse image of a <prime ideal> $\mathfrak p$ under the <Frobenius endomorphism> is
$$
\{a:a^p\in\mathfrak p\}=\mathfrak p,
$$
by primality. The induced continuous map is therefore the identity. These local morphisms agree on overlaps, giving the <Absolute Frobenius morphism> $F_X:X\to X$. Its action on the underlying space and on every local section was prescribed, so the morphism is unique.
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