groupoids versus homotopy 1-types

[John Baez <baez@math.ucr.edu>]

Dear Categorists -

The following claim should be well-known (or false),
but I don't know a reference:

Let Gpd be the 2-category consisting of

groupoids
functors
natural transformations

and let 1Type be the 2-category consisting of

homotopy 1-types
continuous maps
homotopy classes of homotopies

where for present purposes "homotopy 1-types" means "CW complexes with
vanishing higher homotopy groups regardless of the choice of basepoint".

Claim: Gpd and 1Type are equivalent (or "biequivalent",
in older terminology).

In fact I bet there is an explicit pseudo-adjunction between them,
with the "fundamental groupoid" 2-functor going one way and the
"Eilenberg-Mac Lane space" 2-functor going the other way.

Does anyone know for sure?  Know a reference?

Best,
jb