![dual composition](

The one on the left is when you compose it. I think the one in the center is the diagram you drew. Both of these are equal to the right one since all we are doing is extending identities. If you draw it like the diagram on the right, you will be able to separate out the y components and factor out a snake.

There is sliding rule needed to be made rigorous here where you can slide morphisms and stretch the diagrams without changing it but I haven't been able to quite get it.