Hi every one, I have this diagram with isomorphisms, which I can't get to commutate:

the texts says that the horizontal homomorphisms are the obvious ones (whatever is meant by that) and I am to find 3 nonzero vertical homomorphisms so that the diagram commutates. Also to find the kernel-cokernel sequence.

So if I say that

by

by

by

by

and then define

by

by

by

then I can't get the right square to cummutate, for if I take f.eg. z=3 then

How can i get it right?

the kernel would be (I guess?)

because the rows are exact.