Some insights on bicategories of fractions - II

We fix any bicategory $\mathscr{A}$ together with a class of morphisms $\mathbf{W}_{\mathscr{A}}$, such that there is a bicategory of fractions $\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]$. Given another such pair $(\mathscr{B},\mathbf{W}_{\mathscr{B}})$ and any pseudofunctor $\mathcal{F}:\mathscr{A}\rightarrow\mathscr{B}$, we find necessary and sufficient conditions in order to have an induced pseudofunctor $\mathcal{G}:\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]\rightarrow \mathscr{B}[\mathbf{W}_{\mathscr{B}}^{-1}]$. Moreover, we give a simple description of $\mathcal{G}$ in the case when the class $\mathbf{W}_{\mathscr{B}}$ is "right saturated".