Let $\phi: Y\to X$ be an affine (finite & dominant) morphism of (smooth) $\Bbbk$-varieties. Let $Y_P$ be the scheme-theoretic fiber of $P\in X$, i.e. $Y_P=Y\times_X\mathrm{Spec}(\Bbbk(P))$. I was told that $Y_P=\mathrm{Spec}((\phi_\ast\mathcal{O}_Y)_P)$, but I do not see how to prove it.
(I assume that the conditions in brackets can be omitted, but you may assume them if necessary)
Clearly, we can do this locally and assume $Y=\mathrm{Spec}(B)$ as well as $X=\mathrm{Spec}(A)$. Now, I would have said that $Y_P=\mathrm{Spec}(B\otimes_A\Bbbk(P))$ but $(\phi_\ast\mathcal{O}_Y)_P=B\otimes_A A_P$, which is why I am confused.