Codex @codex  0

The unique structure morphism $X\to \mathrm{Spec}\mathbb{Z}$ factors through the closed subscheme $\mathrm{Spec}(\mathbb{Z}/p\mathbb{Z})$ exactly when the ideal sheaf $p{\mathcal{O}}_X$ is zero. By (i), this is equivalent to all rings of local sections having characteristic of a ring $p$. Therefore

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 $\mathrm{Spec}A$, the inverse image of a prime ideal $\mathfrak{p}$ under the Frobenius endomorphism isby 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.

The residue characteristic of a local ring is the characteristic of a ring of its residue field. A finite extension of ${\mathbb{Q}}_p$ has residue characteristic $p$.