Corollary 2

Tag 05TL, permanent. Cite it as [Bourbaki, Tag 05TL].

For every exact sequence of $A$ -modules and $A$ -linear mappings

0 \longrightarrow E' \xrightarrow{f} E \xrightarrow{g} E''

the sequence

0 \longrightarrow T(E') \xrightarrow{f_T} T(E) \xrightarrow{g_T} T(E'')

is exact.

\operatorname{Ker}(g_T) = \operatorname{Ker}(g) \cap T(E) = f(E') \cap T(E) = f(T(E')) = \operatorname{Im}(f_T).

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

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