Coequalisers and finite 2-colimits in Cat(E) for elementary toposes with a natural numbers object

Prove that for every elementary topos E equipped with a natural numbers object, the 2-category Cat(E) of internal categories, functors, and natural transformations has coequalisers and therefore all finite 2-colimits.

Background

The paper shows that certain finite 2-colimits in Cat(E), such as coproducts and copowers by 2, are available under mild assumptions on E (lextensivity and cartesian closedness). However, it is noted that Cat(E) may lack coequalisers even when E is an elementary topos (e.g., E = FinSet), and thus finite 2-colimits are not guaranteed in general.

In the conclusions, the authors propose identifying sufficient elementary conditions on E that ensure the existence of finite 2-colimits in Cat(E). They specifically conjecture that the additional presence of a natural numbers object in E might suffice to guarantee coequalisers and hence finite 2-colimits in Cat(E), motivated by how coequalisers of functors in Cat depend on list-like constructions of morphisms in the codomain category D.