Remark 7.45.4. Consider a commutative diagram

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

of ringed 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*} (h \circ h')^{-1} \circ f_* & = (h')^{-1} \circ h^{-1} \circ f_* \\ & \to (h')^{-1} \circ f'_* \circ g^{-1} \\ & \to f''_* \circ (g')^{-1} \circ g^{-1} \\ & = f”_* \circ (g \circ g')^{-1} \end{align*}

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

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