Lemma 21.39.6. Notation and assumptions as in Example 21.39.1. Let $B \to B'$ be a ring map. Consider the commutative diagram of ringed topoi

Then $L\pi _! \circ Lh^* = Lf^* \circ L\pi '_!$.

Lemma 21.39.6. Notation and assumptions as in Example 21.39.1. Let $B \to B'$ be a ring map. Consider the commutative diagram of ringed topoi

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B}) \ar[d]_\pi & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B'}) \ar[d]^{\pi '} \ar[l]^ h \\ (*, B) & (*, B') \ar[l]_ f } \]

Then $L\pi _! \circ Lh^* = Lf^* \circ L\pi '_!$.

**Proof.**
Both functors are right adjoint to the obvious functor $D(B') \to D(\underline{B})$.
$\square$

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