Let $X$,$Y$ and $Z$ be three toric varieties defined by the fans $\Sigma_X\subset (N_X)_{\mathbb{R}}$, $\Sigma_Y\subset (N_Y)_{\mathbb{R}}$ and $\Sigma_Z\subset (N_Z)_{\mathbb{R}}$, respectively.
It is well known that the product of the two toric varieties $X$ and $Y$ is the toric variety defined by the fan $\Sigma_X\times \Sigma_Y$ in $(N_X)_{\mathbb{R}}\times (N_Y)_{\mathbb{R}}$.
Question: Given two toric morphisms $f:X\to Z$ and $g:Y\to Z$, is there a simple way to describe $X\times_Z Y$?
Thanks a lot !