The connecting homomorphism 
is the homomorphism that,
according to the snake lemma, permits construction
of an exact sequence
 |
(1)
|
from the above commutative diagram whose rows are exact sequences. The homomorphism 
is defined by
 |
(2)
|
for all 
. Here, 
denotes the image, and 
is obtained through the following construction, based on
diagram chasing.
1. Exploit the fact that 
is a surjection to find 
such that 
.
2. Since 
because the right square is a commutative diagram,
belongs to 
,
which is equal to 
due to the exactness
of the lower row at 
. This allows us to find 
such that 
.
While the elements 
and 
are not uniquely determined,
the coset 
is, as can be proven by using more diagram
chasing. In particular, if 
and 
are other elements fulfilling
the requirements of steps (1) and (2), then 
and 
, and
 |
(3)
|
hence 
because of the exactness of the upper row at 
.
Let 
be such that
 |
(4)
|
Then
 |
(5)
|
because the left square is a commutative diagram. Since 
is an injection, it follows that
 |
(6)
|
and so
 |
(7)
|
