Let $X$ be a topological space equipped with maps into two spaces $\bar X_1$ and $\bar X_2$. Is there a standard notation/terminology for the closure $\bar X$ in $\bar X_1 \times \bar X_2$ of the * diagonal map * of $X$?

In my case $X$ is an affine algebraic surface (in fact just $\mathbb{C}^2$) which is isomorphic to Zariski open subsets of complete surfaces $\bar X_1$ and $\bar X_2$ and the maps $X \to \bar X_j$'s are the corresponding embeddings.

In a paper I wrote, I used the notation "birational join" for $\bar X$ following Spivakovsky, but the referee does not like it. (S)He suggested something like the "fiber product" $\bar X_1 \times_X \bar X_2$, but that would require the arrows $X \to \bar X_j$ to be reversed. Similarly the ``cofiber product'' requires the arrows $\bar X \to \bar X_j$ to be reversed.

Any suggestions would be much appreciated. Thanks!