@PiotrAchinger I have a naive question. Does "$B \in \text{Def}_A k[[t]]$ and $C \in \text{Def}_A W(k)$ together define a $D_0 \in \text{Def}_A(R)$" come from the surjectivity of $\text{Def}_A (U \times_V W) \xrightarrow{} \text{Def}_A (U) \times_{\text{Def}_A(V)} \text{Def}_A(W)$? I think the criterion only apply to the case when $U,V, W$ are Artinian and $W$ is a small extension of $V$.

@PiotrAchinge Thank you! Could you tell me why $\text{Def}_A$ being smooth over $W(k)$ will make that works? I am not very familiar with deformation theory.