2-morphisms between geometric morphisms

A two-morphism $f \to g$ is called a geometric transformation. By definition, it is a natural transformation $f^* \Rightarrow g^*$. As mentioned in the nLab article, this is the same thing as a natural transformation $g_* \Rightarrow f_*$.

The convention that it is a natural transformation $f^* \Rightarrow g^*$ (rather than a natural transformation $f_* \Rightarrow g_*$) makes sense if you look at Diaconescu's Theorem. For a presheaf topos $\mathbf{PSh}(\mathcal{C})$ there is an equivalence of categories $$\mathbf{Geom}(\mathcal{E},\mathbf{PSh}(\mathcal{C})) \simeq \mathbf{Flat}(\mathcal{C},\mathcal{E})$$ where $\mathbf{Flat}(\mathcal{C},\mathcal{E})$ denotes the category of flat functors $\mathcal{C} \to \mathcal{E}$ and natural transformations between them. So the direction of the geometric transformation is the same as the direction of the natural transformation between the corresponding flat functors.