  We define weak units in a semi-monoidal 2-category $\CC$ as cancellable
pseudo-idempotents: they are pairs $(I,\alpha)$ where $I$ is an object such
that tensoring with $I$ from either side constitutes a biequivalence of $\CC$,
and $\alpha: I \tensor I \to I$ is an equivalence in $\CC$. We show that this
notion of weak unit has coherence built in: Theorem A: $\alpha$ has a canonical
associator 2-cell, which automatically satisfies the pentagon equation. Theorem
B: every morphism of weak units is automatically compatible with those
associators. Theorem C: the 2-category of weak units is contractible if
non-empty. Finally we show (Theorem E) that the notion of weak unit is
equivalent to the notion obtained from the definition of tricategory: $\alpha$
alone induces the whole family of left and right maps (indexed by the objects),
as well as the whole family of Kelly 2-cells (one for each pair of objects),
satisfying the relevant coherence axioms.
