$require{AMScd}DeclareMathOperatorSpec{Spec}$Let $okay$ be a subject of attribute $p$ and $S$ and $X$ two $okay$-schemes.

Take into account the next diagram the place $X’$ is the pullback of $X$ alongside absolutely the frobenius $f_{Spec okay}$ of S.
$$start{CD}
X’ @>pi_1>> X
@Vpi_2VV @VVfV
Spec okay @>f_{Spec okay}>> Spec okay
finish{CD}$$

Show the next diagram is commutative.
$$start{CD}
X’ instances S @>pi_1 instances f_S>> X instances S
@Vpi_2’VV @VVf’V
S @>f_S>> S
finish{CD}$$

go about this downside?

Limiting to affine schemes $X=Spec R$ and $S=Spec A$ we’d have a hoop diagram:
$$start{CD}
R otimes_{okay, f_k} okay otimes_{A, f_A} A @<pi_1^# otimes f_A<< R otimes A
@Api_2^{prime#}AA @AAf^{prime#}A
A @<f_A<< A
finish{CD}$$

However the way to show it’s commutative? Is $f’^{#}(a) = 1otimes_Aa$ for $ain A$?

Thanks on your assist.