# 2-morphisms between geometric morphisms

Let $$\mathcal E$$ and $$\mathcal F$$ be two toposes. A geometric morphism $$f\colon \mathcal E\to\mathcal F$$ consists of an adjunction $$f^\ast\dashv f_\ast$$.

Let $$f,g\colon \mathcal E\to\mathcal F$$ be two geometric morphisms.

Questions: What is a 2-morphism from $$f$$ to $$g$$ in the 2-category of toposes?

Is it a natural transformation from $$f^\ast$$ to $$g^\ast$$ or from $$f_\ast$$ to $$g_\ast$$?

Can the notion of a 2-morphism from $$f$$ to $$g$$ be defined based on direct image and the inverse image (separately)?

Is a natural transformation from $$f^\ast$$ to $$g^\ast$$ the same as a natural transformation from $$f_\ast$$ to $$g_\ast$$?

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.