Editing Snake lemma (section)

== Construction of the maps ==
[[File:Snake lemma map construction.gif|alt=An animation of the diagram chase to construct the map d by finding d(x) given some x in ker c|thumb|361x361px|An animation of the construction of the map d]]
The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity. The exactness of the two induced sequences follows in a straightforward way from the exactness of the rows of the original diagram. The important statement of the lemma is that a ''connecting homomorphism'' <math>d</math> exists which completes the exact sequence.

In the case of abelian groups or [[module (mathematics)|modules]] over some [[ring (mathematics)|ring]], the map <math>d</math> can be constructed as follows:

Pick an element <math>x</math> in <math>\operatorname{ker} c</math> and view it as an element of <math>C</math>. Since <math>g</math> is [[surjective]], there exists <math>y</math> in <math>B</math> with <math>g(y)=x</math>. By commutativity of the diagram, we have <math>g'(b(y)) = c(g(y)) = c(x) = 0</math> (since <math>x</math> is in the kernel of <math>c</math>), and therefore <math>b(y)</math> is in the kernel of <math>g'</math>. Since the bottom row is exact, we find an element <math>z</math> in <math>A'</math> with <math>f'(z)=b(y)</math>. By injectivity of <math>f'</math>, <math>z</math> is unique. We then define <math>d(x)=z+\operatorname{im}(a)</math>. Now one has to check that <math>d</math> is well-defined (i.e., <math>d(x)</math> only depends on <math>x</math> and not on the choice of <math>y</math>), that it is a homomorphism, and that the resulting long sequence is indeed exact. One may routinely verify the exactness by [[Commutative diagram#Diagram chasing|diagram chasing]] (see the proof of Lemma 9.1 in <ref>{{harvnb|Lang|2002|p=159}}</ref>).

Once that is done, the theorem is proven for abelian groups or modules over a ring.  For the general case, the argument may be rephrased in terms of properties of arrows and cancellation instead of elements.  Alternatively, one may invoke [[Mitchell's embedding theorem]].