[mti]

In that spirit, my pet "a monad is a monoid in the category of endofunctors, duh"-style one-line explanation of the Fourier transform is that it's just the decomposition in the common (Hilbert) eigenbasis for all translation operators. It makes it surprisingly clear (to some) why it is a both natural and important construction.

[franknstein]

Same, but it's only natural after studying inner product vector spaces. Also being comfortable with some calculus is needed to be able to overlook the technicalities of this construction and focus on the actual idea.