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 braidings ${eta}_{r,s}$ in $mathcal{M}$. 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 can define an obvious map ${phi};:;B_g{ ightarrow}{Gamma}_{g,1}$. We show that this map ${phi}$ is injective and nongeometric in the sense of Wajnryb. Since this map extends to a braided monoidal functor ${Phi};:;mathcal{B}{ ightarrow}mathcal{M}$, the integral homology homomorphism induced by ${phi}$ is trivial in the stable range.