Definition 2

Tag 067I, permanent. Cite it as [Bourbaki, Tag 067I].

Algebra, chapter III, § 11. Cogebras, products of multilinear forms, inner products and duality

Given two cogebras E, E', whose coproducts are denoted respectively by c and c', a morphism of E into E' is an A-linear mapping u: E → E' such that (1) $(u \otimes u) \circ c = c' \circ u,$ in other words, it renders commutative the diagram of A-linear mappings

\begin{array}{ccc} E & \xrightarrow{u} & E' \\ \downarrow c & & \downarrow c' \\ E \otimes_A E & \xrightarrow{u \otimes u} & E' \otimes_A E' \end{array}

It is immediately verified that the identity mapping is a morphism, that the composition of two morphisms is a morphism and that every bijective morphism is an isomorphism.

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

Cited by

Example, § 11

Cites

Nothing.