Lemma 91.10.1. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Assume given extensions
as in (91.10.0.1) and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{K} \to \mathcal{L}$.
If there exists an $\mathcal{O}'$-module map $\varphi ' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi $ and $\psi $, then the diagram
\[ \xymatrix{ \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi } & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes _\mathcal {O} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} } \]is commutative.
The set of $\mathcal{O}'$-module maps $\varphi ' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi $ and $\psi $ is, if nonempty, a principal homogeneous space under $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{L})$.
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