I am having trouble understanding the following lemma:

Let $f: M \to N$ be an $A$-module homomorphism. Then there is an exact sequence $$0 \rightarrow K' \stackrel{i}{\rightarrow} M \stackrel{f}{\rightarrow} N \rightarrow C' \stackrel{\pi}{\rightarrow} 0,$$ where $i$ is the inclusion of the kernel of $f$ and $\pi$ is the canonical map onto the cokernel. Conversely, given a six term exact sequence $$0 \rightarrow K' \rightarrow M \stackrel{f}{\rightarrow} N \rightarrow C' \rightarrow 0,$$ $f$ is injective iff $K'=0$ and surjective iff $C'=0$.

I'm having trouble with the converse.First of all, is that six term sequence well-defined? How can it be if there is only one homomorphism involved? Are the homomorphisms suppressed? Should it actually read

$$0 \rightarrow K' \stackrel{g}{\rightarrow} M \stackrel{f}{\rightarrow} N \rightarrow C' \stackrel{h}{\rightarrow} 0?$$ where $g : K' \to M$ and $g : N \to C'$ are homomorphisms. Even with this modification, I'm still having trouble seeing why it is true. Certainly if $K'=0$ and $C'=0$, then $f$ will be bijective. But why should be $K'=C'=0$ if $f$ is bijective?