Is the co-slice category of a pre-topos a pre-topos?

Question

A pre-topos $\mathbb{C}$ is a category which is exact and extensive (it is in particular a coherent category, nLab). Let $X$ be an object of $\mathbb{C}$, and write $\mathbb{C}^{X/}$ for the co-slice category (under-category) of morphisms $X\to A$, $A\in \mathbb{C}$. Is $\mathbb{C}^{X/}$ a pre-topos?

Answer 1

No; here's why: pretoposes always have coproducts; in the co-slice pretopos, coproducts are pushouts in the original pretopos. Pushouts generalize coequalizers, and pretoposes do not generally have coequalizers.