  The disjoint union of mapping class groups of surfaces forms a braided
monoidal category $\mathcal M$, as the disjoint union of the braid groups
$\mathcal B$ does. We give a concrete, and geometric meaning of the braiding
$\beta_{r,s}$ in $\M$. Moreover, we find a set of elements in the mapping class
groups which correspond to the standard generators of the braid groups. Using
this, we obtain an obvious map $\phi:B_g\ra\Gamma_{g,1}$. We show that this map
$\phi$ is injective and nongeometric in the sense of Wajnryb. Since this map
extends to a braided monoidal functor $\Phi : \mathcal B \rightarrow \mathcal
M$, the integral homology homomorphism induced by $\phi$ is trivial in the
stable range.
