Extension of a cylinder functor on C to the S_n C

Question

I was looking at Waldhausen's definition of a cylinder functor and reading his proof that a cylinder functor on $C$ induces cylinder functors on $S_n C$ for all $n$. It seems to me that he is using that a morphism of cofibration sequences $(A \rightarrowtail B \twoheadrightarrow B/A) \to (A' \rightarrowtail B' \twoheadrightarrow B'/A')$ yields a cofibration sequence $T(A \to A') \rightarrowtail T(B \to B') \twoheadrightarrow T(B/A \to B'/A')$. But why should this be true? Or what is he actually doing?