Lemma 18.33.7. Let \mathcal{C} be a site. Let
be a commutative diagram of sheaves of rings on \mathcal{C}. The map \mathcal{O}_2 \to \mathcal{O}'_2 composed with the map \text{d} : \mathcal{O}'_2 \to \Omega _{\mathcal{O}'_2/\mathcal{O}'_1} is a \mathcal{O}_1-derivation. Hence we obtain a canonical map of \mathcal{O}_2-modules \Omega _{\mathcal{O}_2/\mathcal{O}_1} \to \Omega _{\mathcal{O}'_2/\mathcal{O}'_1}. It is uniquely characterized by the property that \text{d}(f) mapsto \text{d}(\varphi (f)) for any local section f of \mathcal{O}_2. In this way \Omega _{-/-} becomes a functor on the category of arrows of sheaves of rings.
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