Lemma 90.18.3. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. Then $f^{-1}L_{\mathcal{B}/\mathcal{A}} = L_{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}$.

**Proof.**
The diagram

\[ \xymatrix{ \mathcal{A}\textit{-Alg} \ar[d]_{f^{-1}} \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar@<1ex>[l] \ar[d]^{f^{-1}} \\ f^{-1}\mathcal{A}\textit{-Alg} \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \ar@<1ex>[l] } \]

commutes. $\square$

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