Corollary
Tag 05P3, permanent. Cite it as [Bourbaki, Tag 05P3].
Algebra, chapter II, § 3. Tensor products
Let F, F', F'' be left A-modules, E a right A-module and
(16) an exact sequence of linear mappings. Writing \bar{s} = lE \otimes s, \bar{t} = l_E \otimes t, the sequence of \mathbf{Z}-homomorphisms*
(17) is exact.
When E (resp. F) is considered as a left (resp. right) A^0-module, F \otimes_{A^0} E is identified with E \otimes_A F and there are analogous identifications for F' \otimes_{A^0} E and F'' \otimes_{A^0} E (no. 1, Corollary 2 to Proposition 1); the corollary then follows immediately from Proposition 5.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.