A homomorphism
f : M > N is represented as a matrix from the generators of M to the generators of N.
i1 : R = QQ[x,y]/(y^2x^3);

i2 : M = module ideal(x,y)
o2 = image  x y 
1
o2 : Rmodule, submodule of R

One homomorphism
F : M > R is
x > y, y > x^2 (this is multiplication by the fraction
y/x). We write this in the following way.
i3 : F = map(R^1,M,matrix{{y,x^2}})
o3 =  y x2 
1
o3 : Matrix R < M

Notice that as is usual in Macaulay2, the target comes before the source.
Macaulay2 doesn't display the source and target, unless they are both free modules or have been assigned to global variables. Use
target and
source to get them. The
matrix routine recovers the matrix of free modules between the generators of the source and target.
i4 : source F
o4 = image  x y 
1
o4 : Rmodule, submodule of R

i5 : target F == R^1
o5 = true

i6 : matrix F
o6 =  y x2 
1 2
o6 : Matrix R < R

Macaulay2 also does not check that the homomorphism is well defined (i.e. the relations of the source map into the relations of the target). Use
isWellDefined to check. This generally requires a Gröbner basis computation (which is performed automatically, if it is required and has not already been done).
i7 : isWellDefined F
o7 = true

i8 : isIsomorphism F
o8 = false

The image of
F lies in the submodule
M of
R^1. Suppose we wish to define this new map
G : M > M. How does one do this?
To obtain the map
M > M, we use
Matrix // Matrix. In order to do this, we need the inclusion map of
M into
R^1.
i9 : inc = inducedMap(R^1, M)
o9 =  x y 
1
o9 : Matrix R < M

Now we use // to lift
F : M > R^1 along
inc : M > R^1, to obtain a map
G : M > M, such that
inc * G == F.
i10 : G = F // inc
o10 = {1}  0 x 
{1}  1 0 
o10 : Matrix M < M

i11 : target G == M and source G == M
o11 = true

i12 : inc * G == F
o12 = true

Let's make sure that this map
G is well defined.
i13 : isWellDefined G
o13 = true

i14 : isIsomorphism G
o14 = false

i15 : prune coker G
o15 = cokernel  y x 
1
o15 : Rmodule, quotient of R

i16 : kernel G == 0
o16 = true
