Example
Tag 056W, permanent. Cite it as [Bourbaki, Tag 056W].
Algebra, chapter I, § 2. Identity element; cancellable elements; invertible elements
(1) An identity element is its own inverse.
(2) In the set of mappings of E into E an element f is left invertible (resp. right invertible) if f is a surjection (resp. injection). The left inverses (resp. right inverses) are then the retractions (resp. sections) associated with f (Set Theory, II, § 3, no. 8, Definition 11). For f to be invertible it is necessary and sufficient that f be a bijection. Its unique inverse is then the inverse bijection of f.
Let E and F be two unital magmas and f a unital homomorphism of E into F. If x' is the inverse of x in E, f(x') is the inverse of f(x) in F. Hence, if x is an invertible element of E, f(x) is an invertible element of F.
In particular, if R is an equivalence relation compatible with the law on a unital magma E, the canonical image of an invertible element of E is invertible in E/R.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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