Proposition 11

Tag 06D8, permanent. Cite it as [Bourbaki, Tag 06D8].

— If $M$ is free, then $\varphi_M$ is a morphism of graded bigebras.

Using the commutativity of the diagrams (13) and (14), we obtain the commutative diagram

\begin{array}{ccccccccc} S(M) & \xrightarrow{S(\Delta)} & S(M \oplus M) & \xrightarrow{h} & S(M) \otimes S(M) \\ \downarrow \varphi_M & & \downarrow \varphi_{M \oplus M} & & \downarrow \varphi_M \otimes \varphi_M \\ TS(M) & \xrightarrow{TS(\Delta)} & TS(M \oplus M) & \xrightarrow{k} & TS(M) \otimes TS(M), \end{array}

where $\Delta$ is the diagonal homomorphism and $h, k$ are canonical homomorphisms. The proposition follows from this.

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

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