Let $C$ and $D$ be sites. Each morphism of sites $f : C \rightarrow D$ gives an essential geometric morphism $f_! \dashv f^* \dashv f_* : [C^{op}, \text{Set}]_{\text{Cat}} \rightarrow [D^{op}, \text{Set}]_{\text{Cat}}$.
We also get an adjunction $f_* : \text{Sh}(C) \rightarrow \text{Sh}(D): f^*$, but not necessarily a $f_!$ here. From what I understand, geometric morphisms arose when people tried to encapsulate the properties of $f_*$ and $f^*$ here; the adjunction $f_* \dashv f^*$ always has the extra property that $f^*$ preserves finite limits. Further, embeddings are characterized as embeddings of sheaf categories into their presheaf categories, up to natural isomorphism.
On the other hand, we could try to synthesize this setup in another way: there are geometric embeddings $\text{Sh}(C) \rightarrow [C^{op}, \text{Set}]$ and $\text{Sh}(D) \rightarrow [D^{op}, \text{Set}]$, making a commutative diagram:
In this diagram, the top adjunction is an essential geometric morphism, and the vertical adjunctioins are geometric embeddings. The right adjoints of make a commutative diagram, so that the left adjoints commute up to natural isomorphism. Further, the top and vertical left adjoinits preserve finite limits, from which we can see that the bottom does as well.
My question is, could we equally well define a geometric morphism of grothendieck topoi to be one which arises from a setup such as this?