As stated in [this](https://mathoverflow.net/questions/89444/do-disjoint-unions-and-fiber-products-commute) question, coproducts commute with pullbacks in the category of sets.


Let $Grpd\_{\infty}$ denote the $\infty$-category of $\infty$-groupoids. Do coproducts commute with pullbacks in $Grpd\_{\infty}$?



Do coproducts of infinity-groupoids commute with pullbacks?
Yes, they do, if you mean what is meant in the question you linked, but it is misleading to describe this property as "coproducts commute with pullbacks", say instead "pullback along a fixed morphism preserves coproducts", or "pullback *functors* preserve coproducts", or with more jargon, "coproducts are universal".


More generally, colimits are universal and this is one of the defining properties of an $\infty$-topos. See the discussion in section 6.1 of Lurie's *Higher Topos Theory*, specifically proposition 6.1.3.10.




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EDIT: Thanks to Marc Hoyois for pointing out an absent minded mistake in a previous version!


The thing I would call "coproducts commuting with pullbacks", is the following property: given diagrams $X\_i \to Z\_i \leftarrow Y\_i$, the canonical morphism $\coprod\_i X\_i \times\_{Z\_i} Y\_i \to (\coprod\_i X\_i) \times\_{\coprod\_i Z\_i} \coprod\_i Y\_i$ is an isomorphism (or equivalence, if you prefer that term). This also holds in spaces and indeed in every $\infty$-topos. More generally, in $\infty$-toposes coproducts commute with limits over connected categories.


