  Given a locally compact group $G$, we study the smallest exact
crossed-product functor $(A,G,\alpha)\mapsto A\rtimes_{\mathcal E} G$ on the
category of $G$-$C^*$-dynamical systems. As an outcome, we show that the
smallest exact crossed-product functor is automatically Morita compatible, and
hence coincides with the functor $\rtimes_{\mathcal{E}}$ as introduced by Baum,
Guentner, and Willett in their reformulation of the Baum-Connes conjecture (see
[2]). We show that the corresponding group algebra $C_{\mathcal{E}}^*(G)$
always coincides with the reduced group algebra, thus showing that the new
formulation of the Baum-Connes conjecture coincides with the classical one in
the case of trivial coefficients.
  Erratum: After publication of this manuscript, some gaps have unfortunately
been found affecting some parts of the paper. We therefore included an appendix
with an erratum at the end of this paper explaining the mistakes and keeping
the original published version unchanged.
