Total Products

Let $\mathcal{E}\to\mathcal{B}$ be a Displayed category with $\mathcal{B}$ Cartesian. We say that $\mathcal{E}$ has total products if the Total Category $\int\mathcal{E}$ is itself Cartesian.

In displayed terms, this means that for $a,b:\mathcal{B}$ and $x:\mathcal{E}_{a},y:\mathcal{E}_{b}$, we have a product $x\times y:\mathcal{E}_{a\times b}$ where the projections are appropriately displayed over the projections in $\mathcal{B}$.