Seifert-Van Kampen theorem

Statement

Let $X$ be a topological space and let $𝒰=\{U_i{\}}_{i\in I}$ be an open cover of $X$.

Let $ℱ$ be the set of all finite non-empty intersections of members of $𝒰$: $$ℱ=\{\bigcap _{i\in J}{U}_i|\mathrm{\varnothing }\ne J\subseteq I,|J|<\mathrm{\infty }\}$$.

Regard $ℱ$ as a category whose objects are the elements of $ℱ$ and in which there is a unique morphism $V\to W$ whenever $V\subseteq W$.

Let ${\Pi }_1:ℱ\to 𝖦𝗉𝖽$ be the functor that sends each $V\in ℱ$ to its fundamental groupoid ${\Pi }_1(V)$ and each inclusion $V\hookrightarrow W$ to the induced morphism of groupoids.

Then the canonical morphism $${\mathrm{colim}}_{V\in ℱ}{\Pi }_1(V)\to {\Pi }_1(X)$$ is an isomorphism of groupoids.