First, a Morita map (also known as a weak equivalence or essential equivalence) between two Lie groupoids G 1 ⇉ ⇉ G 0 and H 1 ⇉ ⇉ H 0 consists of a Lie groupoid morphism from G to H which is moreover fully faithful and essentially surjective (adapting these categorical notions to the smooth context). We say that two Lie groupoids G 1 ⇉ ⇉ G 0 and H 1 ⇉ ⇉ H 0 are Morita equivalent if and only if there exists a third Lie groupoid K 1 ⇉ ⇉ K 0 together with two Morita maps from G to K and from H to K.