Let $A$ be a characteristic $p>0$ commutative ring. Let $B$ be a finitely presented etale $A$ algebra i.e. 
$$
B=A[x_1,\ldots,x_n]/(f_1,\ldots,f_n),
$$
with $det(\frac{\partial f_i}{\partial x_j})\in B^{\times}$. Consider the multiplication map
\begin{align*}
m:A\otimes_A B\rightarrow B\\\
      a\otimes b\mapsto ab^p
\end{align*}
Note here that for $a\in A$ we have the tensor relation $1\otimes ab=a^p\otimes b$.

Q: How does one prove directly that $m$ is an isomorphism of rings?