An $n$-fold category is an internal category in the category of $(n-1)$-fold categories (and a $0$-fold category is just a Set).

General results about internal categories assure that the category of $n$-fold categories is cartesian closed. Is something known about colimits of $n$-fold categories?

Intuitively, finite coproducts exist. What about coequalizers?

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    $\begingroup$ Colimits in the strict sense exist, for general reasons. What would be more interesting is non-strict colimits. $\endgroup$ – Zhen Lin Oct 6 '15 at 14:04
  • $\begingroup$ @ZhenLin Can I ask you what those general reasons are? $\endgroup$ – Maxime Lucas Oct 6 '15 at 14:33
  • $\begingroup$ The category of internal categories in a locally presentable category is again a locally presentable category. In particular, it is has colimits. $\endgroup$ – Zhen Lin Oct 6 '15 at 16:08
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    $\begingroup$ Following the (external construcion) in Wolff, H. V-cat and V-graph, J. Pure Appl. Algebra 4 (1974), 123—135, rewrite this proof in internal terms... $\endgroup$ – Buschi Sergio Oct 6 '15 at 16:52

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