A characterization of the relation between two $\ell$-modular correspondences

Let $F$ be a non archimedean local field of residual characteristic $p$ and $\ell$ a prime number different from $p$. Let $\mathrm{V}$ denote Vign\'eras' $\ell$-modular local Langlands correspondence between irreducible $\ell$-modular representations of $\mathrm{GL}_n(F)$ and $n$-dimensional $\ell$-modular Deligne representations of the Weil group $\mathrm{W}_F$. In a previous work, enlarging the space of parameters to Deligne representations with non necessarily nilpotent operators, we proposed a modification of the correspondence of Vign\'eras into a correspondence $\mathrm{C}$ compatible with the formation of local constants in the generic case. In this note, following a remark of Alberto M\'inguez, we characterize the modification $\mathrm{C}\circ \mathrm{V}^{-1}$ by a short list of natural properties.