TITLE: A connected groupoid is equivalent to a groupoid with one object
QUESTION [5 upvotes]: This is same as saying a connected groupoid is equivalent to a group. But I have no idea how to construct such a group, can't even get started.
Any help is appreciated.

REPLY [4 votes]: Using a different terminology, but basically similar reasoning as Dori:
Every category is equivalent to its skeleton
Every connected groupoid is a category
In a skeleton you fuse together all isomorphic objects, therefore the skeleton of a connected groupoid is a category with a single element and invertible morphisms (a group)
So every connected groupoid is equivalent to a group