The present work describes the explicit calculation of mixed concomitants for $G = GL (k,\mathbf{C})$ and their relations. A set of generators for the module $\mathbf{C} \left[ S_{m}\right]^{Hom_{G}((V \otimes V^{\star})^{\otimes m} , (V \otimes V^{\star})^{\otimes n})}\mathbf{C} \left[ S_{n}\right]$ is also exhibited. In view of the results of [1], everything remains true for any field of characteristic $p\ge 0$.