I know what the frobenius map is the naive sense, just the $p$th power map on an $\mathbb{F}_p$ algebra, and that this can be upgraded somewhat to the generator of the monoid of natural transformations on the identity functor on $\mathbb{F}_p$ algebras. To my knowledge, this is the reason it pops up anywhere one has anything of/over characteristic $p$.
For instance, the frobenius morphism I have seen in algebraic geometry seem to just be unwinding the definitions down until we get elements of an $\mathbb{F}_p$ algebra, then applying this down to earth frobenius map.
However, this doesn't seem fully satisfactory, in that before doing any of this, we fix a prime $p$, and use the frobenius map in this fixed characteristic.
So my question is this:
In what sense are the frobenius maps for different $p$ the same thing, can they be viewed as manifestations of the same underlying morphism of something?