nLab uses the following definition of van Kampen colimits --- a colimit in a category $\mathbb{C}$ is called van Kampen iff it is preserved by the internal indexing functor $\mathbb{C}/(-) \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$ defined as: $$X \mapsto \mathbb{C}/X$$ $$X \overset{f}\rightarrow Y \mapsto \mathbb{C}/Y \overset{f^*}\rightarrow \mathbb{C}/X$$ where $f^*$ is the pullback-along-$f$ functor.

My question is --- where does this definition come from and why are such colimits called "van Kampen"?

In case of coproducts one may notice that the property of being van Kampen in the above sense is equivalent to the usual property of being extensive.

On the other hand, van Kampen pushouts in the above sense do not match the usual definition of van Kampen pushouts from the definition of an adhesive category. For example, if $\mathbb{C} = \mathbf{Set}$ then the internal indexing functor $\mathbf{Set}/(-)$ is equivalent to the usual exponential functor $\mathbf{Set}^{(-)}$, which (like every exponential functor) preserves all colimits; but not every pushout in $\mathbf{Set}$ is van Kampen (pushouts along monomorphisms are).

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