This is a basic question about base change of projective cones, whose support may not be the entire base scheme.

Let $X$ be a scheme of finite type over a field, and $S^\cdot$ a sheaf of graded $\mathcal{O}_X$ algebras on $X$ generated by $S^1$ over $S^0$, with the natural map $\mathcal{O}_X \rightarrow S^0$ surjective. $P = \textrm{Proj}(S^\cdot)$, along with it's nautral projection $\pi$ to $X$, is a projective cone on $X$ with support defined by the ideal sheaf $\ker( \mathcal{O}_X \rightarrow S^0)$.

Let $i: Z \rightarrow X$ be a closed immersion. Base change $\pi$ along $i$. Then, do we have:

1.) The fiber product is the cone corresponding to the pullback $i^*(S^\cdot)$ sheaf of algebras on $Z$.

2.) The induced morphism on cones pulls back $O_P(1)$ to $O_Z(1)$ where these sheaves denote the canonical sheaves on the projective cones over $X$ and $Z$ respectively.

In his intersection theory book, Fulton only states (1,2) as true in a more restrictive setting ($i$ can be replaced by an arbitrary proper morphism but $S^\cdot$ must correspond to a vector bundle).

He states (1) not for projective cones, but for cones (replace $\textrm{Proj}$ above with $\textrm{Spec}$) that moreover have the natural map $\mathcal{O}_X \rightarrow S^0$ an isomorphism (even though he gives the more general definition above). To work out the details of something in his book I'd like this stronger result above. I'm asking for a reference in case such a result is actually false in this stronger setting.