Remark

(1) If the ordered pair $(f, g)$ is an exact sequence, then $g \circ f = 0$; but of course this property does not characterize exact sequences, for it only means that the image of $f$ is contained in the kernel of $g$.

In the statements below, E, F, G denote A-modules, 0 the A-module reduced to its identity element; the arrows represent A-module homomorphisms. As there is only one homomorphism from the module 0 to a module E (resp. of E to 0), there is no point in giving these homomorphisms a name in the exact sequences where they appear.