Corollary 1

Suppose that the diagram (10) is commutative and the two rows are exact. Then:

(i) If $u'$, a and c are injective, b is injective. (ii) If v, a and c are surjective, b is surjective.

Assertion (i) is a consequence of assertion (i) of Proposition 2: for $\mathrm{Ker}(a) = 0$ and $\mathrm{Ker}(c) = 0$, hence $\mathrm{Ker}(b) = 0$.

Assertion (ii) is a consequence of assertion (ii) of Proposition 2: for $\mathrm{Coker}(a) = 0$ and $\mathrm{Coker}(c) = 0$, hence $\mathrm{Coker}(b) = 0$.