Prove $\mathcal{O}_{X_{\text{ét}}}(U) = \Gamma(U,\mathcal{O}_{U})$ is a sheaf on $X_{\text{ét}}$.

For $U \to X$ étale, define a presheaf by $\mathcal{O}_{X_{\text{ét}}}(U) = \Gamma(U,\mathcal{O}_{U})$. I want to show that this is a sheaf on $X_{\text{ét}}$.

Clearly, the sheaf condition holds for all Zariski open coverings, so it is sufficient to show the sheaf condition holds for étale coverings of the form $\{\operatorname{Spec}B \to \operatorname{Spec}A\}$ consisting of a single surjective étale morphism.

This leads us to check the sequence $0 \to A \to B \to B \otimes_{A} B$ is exact where $A \to B$ is the morphism induced by $\operatorname{Spec}B \to \operatorname{Spec}A$ and the map $B \to B \otimes_{A} B$ is the map $b \mapsto 1 \otimes b - b \otimes 1$.

The map $A \to B$ is injective as $\operatorname{Spec}B \to \operatorname{Spec}A$ is surjective. The problem I have is showing that $\operatorname{im} (A \to B) = \ker (B \to B \otimes_{A} B)$.

There is a proof in Milne, but I'm using a different definition of étale and I am not using faithfully flat morphisms. I'm using the definition of étale found in Vakil (smooth of relative dimension $0$), also shown in this question here.