Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$.

Let $\Gamma$ be the graph of the inverse of geometric Frobenius $\phi\otimes\overline{\mathbf{F}}_p$, with $\phi$ the $p$-th power map on $\mathbf{P}^1_{\mathbf{F}_p}$.

The intersection $\Gamma\cap \Delta$, with $\Delta$ the diagonal, is $X(\mathbf{F}_p)$, although intuitively each of the elements of $X(\mathbf{F}_p)$ should be counted with multiplicity $p$.

Is this intuition correct and is there a way to make it precise?

I would be interested in a reference, if any.