Remark 21.19.4. Consider a commutative diagram

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'), \mathcal{O}_{\mathcal{B}'}) \ar[r]_ k \ar[d]_{f'} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B}) \ar[d]^ f \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]^ l \ar[d]_{g'} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[d]^ g \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^ m & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \\ } \]

of ringed topoi. Then the base change maps of Remark 21.19.3 for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition

\begin{align*} Lm^* \circ R(g \circ f)_* & = Lm^* \circ Rg_* \circ Rf_* \\ & \to Rg'_* \circ Ll^* \circ Rf_* \\ & \to Rg'_* \circ Rf'_* \circ Lk^* \\ & = R(g' \circ f')_* \circ Lk^* \end{align*}

is the base change map for the rectangle. We omit the verification.

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