The Stacks project

Definition 18.8.1. Let $f, g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be two morphisms of ringed topoi. A 2-morphism from $f$ to $g$ is given by a transformation of functors $t : f_* \to g_*$ such that

\[ \xymatrix{ & \mathcal{O}_\mathcal {D} \ar[ld]_{f^\sharp } \ar[rd]^{g^\sharp } \\ f_*\mathcal{O}_\mathcal {C} \ar[rr]^ t & & g_*\mathcal{O}_\mathcal {C} } \]

is commutative.

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View Definition 18.8.1 as pdf