Corollary

Tag 01YV, permanent. Cite it as [Bourbaki, Tag 01YV].

Topologie algébrique, chapter III, § 3. Groupoïde de Poincaré

Let $x= (x_j)_{j\in J}$ be a point of X. The mapping

\pi_1(X, x)\rightarrow \prod_{j\in J}\pi_1(X_j, x_j)

deduced from the mappings $\pi_1$ (pr $_j, x_j$ ) is a group isomorphism.

This isomorphism is said to be canonical. In the sequel, we shall often identify $\pi_1(X, x)$ with $\prod_{j\in J}\pi_1(X_j, x_j)$ by means of this isomorphism.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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