Lie groupoids are groupoids with smooth structures. There is a nature 2-category of Lie groupoids: Lie groupoids, smooth functors of Lie groupoids, smooth natural transformations of smooth functors. However this is usually not the correct one, partially because weak equivalence (fully-faithful and essential surjective) of Lie groupoids cannot be inverted. We need more morphisms between Lie groupoids. There are three (or 2.5) ways to define the more morphisms:

1) Span: Span of Lie groupoid morphism G<<--K-->H such that the left leg is a weak equivalence. 
   Equivalence of such spans: G<<--K1-->H and G<<--K2-->H  are equivalent if there exists G<<--K3-->H with K1<<--K3-->>K2 such that all triangles 2-commute.

1') Span:Span of Lie groupoid morphism G<<--K-->H such that the left leg is a weak equivalence and the map between objects are surjective submersion. This is a slightly different version of the 1). 

2) Bibundle: right principal bibundle of Lie groupoids.
   Equivalence of bibundles: equivariant map of bibundles (this is in fact a diffeomorphism).

Modulo equivalent relation in three cases we obtain three categories. A classic result tell us that they are isomorphic.

One could go a bit further, it is possible to define three 2-categories (i.e. bi-categories, in fact (2,1)-categories, 2-morphisms are invertible), and naturally there are functors between them. The construction is given in Hellen Colman http://www.springerlink.com/content/3472617rj6178271/.

One expect also that they are 2-equivalent 2-categories. Recall that a equivalent functor of 2-categories must be locally equivalent and surjecttive-up-to-equivalence on objects (See Leinster "Basic Bicategories"). This means that 2-morphisms must be the same. However it seem that 2-morphisms of (1) and (2) are different. The 2-morphisms induced from (2) to (1) must be and strict isomorphism (namely, K1 and K2 are isomorphic). What's wrong with my reasoning?