# When do pushouts along epis preserve products?

A pushout diagram in a category $\mathcal{C}$ is a commutative square with a certain universal property; as usual, say that the pushout diagram is along an epi if at least one of the two arrows out of its source object is an epi.

Assuming $\mathcal{C}$ has products, then given two pushout diagrams $D_1,D_2$, their product is a commutative square, but not generally a pushout diagram. What if we additionally assume that $D_1$ and $D_2$ are each along an epi?

Of course, we do not expect the result to be a pushout diagram in an arbitrary $\mathcal{C}$, but are there nice cases where we do expect this?

Question: Is there any well-known sort of category $\mathcal{C}$ for which pushout-along-epis commutes with finite products?

A special case of the above question was considered in this MO question, where Todd Trimble showed that it holds.

• Doesn't Todd's argument work in any $\infty$-pretopos? – Mike Shulman Jun 22 '16 at 21:21