Recall that "transpose" is not only defined for square matrices. If A is an n by m matrix and B is an m by p matrix, so that you can multiply them, then AT is an m by n matrix and BT is a p by m matrix. If n is not equal to p, you can't multiply ATBT. But you can multiply BTAT.
if you know what it means in terms of being the induced map, namely composition, on dual spaces, the result is just the obvious fact that f*g*(h) = f*(g*h) = f*(hog) =
hogof = (gof)*(h) = (gf)*(h), so (gf)* = f*g*.
what you need to do is to do a matrix multiplication for an abitrary matrix A and matrix B, I should say a transpose multiplication of matrix A and then transpose of matrix B. Then find the multiplication of matrix AB and find the transpose of that. It should be the same and that should do it.