Remark 2

Tag 0BIS, permanent. Cite it as [Bourbaki, Tag 0BIS].

General Topology, chapter III, § 2. Subgroups, quotient groups, homomorphisms homogeneous spaces, product groups

Let $f$ be a strict morphism of G into $G'$ and let $g$ be a strict morphism of $G'$ into $G''$ . If $f$ is surjective or if $g$ is injective, it follows immediately from Proposition 24 that $g \circ f$ is a strict morphism of G into $G''$ . But this conclusion is no longer necessarily valid if neither of the two preceding conditions is satisfied, even if $f$ is injective and $g$ surjective (Exercise 19).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Lemma 2, § 2

Cites

Exercise 19
Proposition 24