Situation 91.9.1. A morphism of thickenings of ringed topoi (f, f') is given by a commutative diagram
of ringed topoi whose horizontal arrows are thickenings. In this situation we set \mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp ) \subset \mathcal{O}' and \mathcal{J} = \mathop{\mathrm{Ker}}(t^\sharp ) \subset \mathcal{O}_{\mathcal{B}'}. As f = f' on underlying topoi we will identify the pullback functors f^{-1} and (f')^{-1}. Observe that (f')^\sharp : f^{-1}\mathcal{O}_{\mathcal{B}'} \to \mathcal{O}' induces in particular a map f^{-1}\mathcal{J} \to \mathcal{I} and therefore a map of \mathcal{O}'-modules
If i and t are first order thickenings, then (f')^*\mathcal{J} = f^*\mathcal{J} and the map above becomes a map f^*\mathcal{J} \to \mathcal{I}.
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