Then the induced maps on homology form a commutative morphism between the associated long exact homology sequences. Explicitly, for every $n \in \mathbb{Z}$, the squares
commute, where $\partial_n: H_n(C_*) \to H_{n-1}(A_*)$ and $\partial'_n: H_n(C'_*) \to H_{n-1}(A'_*)$ are the connecting homomorphisms of the two short exact sequences.