Corollary

For an A-linear mapping $u : E \to F$ to be surjective (resp. bijective, resp. zero), it is necessary and sufficient that, for every A-module G, the mapping $\operatorname{Hom}(u, 1_G) : \operatorname{Hom}(F, G) \to \operatorname{Hom}(E, G)$ be injective (resp. bijective, resp. zero).

It suffices to apply Theorem 1 to the case where $E'' = \{0\}$ (resp. $E' = \{0\}$ resp. $E'' = E$ and $v = 1_E$).

Note that starting from an exact sequence

the corresponding sequence

is not necessarily exact, in other words, the homomorphism $\bar{u}$ is not necessarily surjective. If $E'$ is identified with a submodule of E, this means that a linear mapping of $E'$ into F cannot always be extended to a linear mapping of E into F (Exercises 11 and 12). However: