Lemma 4

Tag 0DW7, permanent. Cite it as [Bourbaki, Tag 0DW7].

If $B(C)$ and $B(C')$ are projective, then the canonical homomorphism

\gamma(C, C') : H(C) \otimes_A H(C') \to H(C \otimes_A C')

has a $k$ -linear retraction. Indeed by X, p. 65, remark b), there exist homologisms

\varphi : C \to H(C)

and

\varphi' : C' \to H(C')

such that

H(\varphi) = 1_{H(C)}

and

H(\varphi') = 1_{H(C')}

. In the commutative diagram

\begin{array}{ccc} H(C) \otimes_A H(C') & \xrightarrow{\gamma(C, C')} & H(C \otimes_A C') \\ \downarrow H(\varphi) \otimes H(\varphi') & & \downarrow H(\varphi \otimes \varphi') \\ H(C) \otimes_A H(C') & \xrightarrow{\gamma(H(C), H(C'))} & H(C) \otimes_A H(C') \end{array}

$H(\varphi) \otimes H(\varphi')$ and $\gamma(H(C), H(C'))$ are the identity, whence the assertion.

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

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