On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra image of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra image which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over image we construct the associated spectral étale Lie groupoid image over M such that image is naturally isomorphic to G. Both these constructions are functorial, and image is fully faithful left adjoint to image. We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid image of an étale Lie groupoid G.