Exactness of $A\to B\to C$

A sequence of homomorphisms \(A \mapsvia{f_1} B \mapsvia{f_2} C\) is exact at \(B\) if and only if \(\im f_1 = \ker f_2\). In that case \(f_2 \circ f_1 = 0\); the converse fails, since \(f_2\circ f_1 = 0\) only gives \(\im f_1\subseteq\ker f_2\).