Composition Of Functions Implies Equality

[netcaster]

We have three functions: f:A->A, g:A->A and h:A->A
with both f and g bijective and h bijective.

We know that f ° h = h ° g for every x in A.

Is it true that f=g for every x in A?

I have tried to solve it and I am pretty sure it is true but I can find neither a counterexample nor a simple proof.

 

[HallsofIvy]

Take A= {1, 2, 3}, f= {(1,3), (2,1), (3,2)}, g= {(1,2), (2,3), (3,1)}, h= {(1,3), (2,3), (3, 1)}.