The following definition of a deformation of a scheme is taken from Grothendieck's FGA explained:Fantechi.
Let $i:S_0 \to S$ be a thickening of order one defined by an ideal $I$ of square zero and let $X_0$ be a flat sch eme over $S_0$. By a deformation of $X_0$ over $S$ we mean a cartesian square,
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with $X$ flat over $S$. The flatness condition is expressed by the fact that the natural map $f_0^*I \to J$, where $f_0:X_0 \to S_0$ is the structural morphism and $J$ is the ideal of $X_0$ in $X$, is an isomorphism.
I am really confused as to how exactly $I$ sits inside of $S_0$. We know that $I$ is a sheaf of ideals of $S$, and similarly $J$ is a sheaf of ideals in $X$, however, how do $I$ and $J$ sit inside of $S_0$ and $X_0$, respectively?
The morphism $f_0^*I \to J$ really doesn't make sense to me since to me since it seems like both $I$ and $J$ would vanish as ideals on $X_0$ and $S_0$.
Can somebody explain me what exactly is meant by the morphism $f_0^*I \to J$? In particular, how to $I$ and $J$ sit in $X_0$ and $S_0$?
$\textbf{My thoughts:}$ I thought maybe what is meant by this morphism is the morphism $\mathcal{O}_S/I \to (f_0)_* \mathcal{O}_X/J$ and then take the pushforward $f_0^* \mathcal{O}_S/I \to f_0^*(f_0)_* \mathcal{O}_X/J=\mathcal{O}_X/J$. However, if I look a bit further on in the book and take a look at obstruction theory, I realize that it really doesn't make sense for $f_0^*I$ to denote $f_0^*\mathcal{O}_S/I$.