Remark 7.45.3. Consider a commutative diagram

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

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

\begin{align*} m^{-1} \circ (g \circ f)_* & = m^{-1} \circ g_* \circ f_* \\ & \to g'_* \circ l^{-1} \circ f_* \\ & \to g'_* \circ f'_* \circ k^{-1} \\ & = (g' \circ f')_* \circ k^{-1} \end{align*}

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

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