Let $f: X \to Y= \operatorname{Spec A}$ be a separated morphism of finite type of noetherian schemes and let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. For a point $y \in Y$, let $X_y$ be the fiber over $y$ and let $\mathcal{F}_y$ be the induced sheaf.
Let $i: X_y \to X$. Is $\mathcal{F}_y$ the restriction of $\mathcal{F}$ onto the fiber $X_y$, i.e $\mathcal{F}_y = i^{-1} \mathcal{F}$, which is not necessarily locally free or even an $\mathcal{O}_X$-module, or is $\mathcal{F}_y$ the pullback, $i^* \mathcal{F}$ which is locally free and an $\mathcal{O}_x$-module?
I imagine $\mathcal{F}_y$ should be the latter but I am not exactly sure what Hartshorne has in mind here.
Furthermore, in the same corollary, he goes on to define the following sheaf: $\mathcal{F} \otimes k(y)$ where $k(y)$ is the constant sheaf on the closed subset $\overline{ \{ y \} }$ of $Y$.
First of all, is $k(y)$ with this definition a sheaf on $Y$ via extension by zero?
Also, I thought to take the tensor product of two sheaves they must both be modules over the same structure sheaf. However when Hartshorne writes $\mathcal{F} \otimes k(y)$ he never designates what sheaf we are tensoring over.