  We show that every braided monoidal category arises as $\End(I)$ for a weak
unit $I$ in an otherwise completely strict monoidal 2-category. This implies a
version of Simpson's weak-unit conjecture in dimension 3, namely that
one-object 3-groupoids that are strict in all respects, except that the object
has only weak identity arrows, can model all connected, simply connected
homotopy 3-types. The proof has a clear intuitive content and relies on a
geometrical argument with string diagrams and configuration spaces.
