Consider the following cartesian diagram of schemes: $$\begin{array} AX^{'} & \stackrel{v}{\longrightarrow} & X \\ \downarrow{u} & & \downarrow{f} \\ \mathrm{Spec}A & \stackrel{g}{\longrightarrow} & Y \ . \end{array} $$ Let $F$ be a sheaf on $X$. Then It seems that $F \otimes A$ is used in many books of algebraic geometry instead of $v^*F$ . Does $F \otimes A$ means $u^*(f_*F \otimes _ {O_Y} A^{\sim})$ ? In this case, is it canonically isomorphic to $v^*F$ ?
I see that $v^*F$ is isomorphic to $F \otimes A$ in affine case.