A crossconnections between two balanced categories $\mathcal{C}$ and $\mathcal{D}$ is a local isomorphism $\Gamma : \mathcal{D}\to B*\mathcal{C}$ where $B*\mathcal{C}$ is the balanced dual of $\mathcal{C}$ such that the image of $\Gamma$ is total in $B*\mathcal{C}$. It is also such a crossconnection $\Gamma$ determines a concordant semigroup.