Corollary 2
Tag 05DL, permanent. Cite it as [Bourbaki, Tag 05DL].
Algebra, chapter I, § 6. Extensions, solvable groups, nilpotent groups
Let f be a homomorphism from a nilpotent group G to a group G'. If the restriction of f to the centre of G is injective, f is injective.
This is Corollary 1 applied to Ker(f).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.