Question

I found the following argument in more than one article: Let $X_0$ be a complex space and define the functor $F:Art \to Sets$ s.t. $F(A)={\text{Isomorphism classes of deformations of X_0 over A}}$. Let $(X",A"),(X',A'), (X,A)$ deformations of $X$ and consider 2 maps $A"\to A$ and $A' \to A$ such that the first one is surjective, so we can consider $A' \times_{A} A"$. Then a deformation over $A' \times_{A} A"$ is $Y=(X_0,\mathcal{O}_{X'} \times_{\mathcal{O}_X} \mathcal{O}_{X"})$. Can someone explain to me what is Y? Suppose that $X=V(f)$ for some $f\in \mathbb{C}[x_1, \dots, x_n]$, how I compute Y?