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Remark 20.28.4. Consider a commutative diagram

\[ \xymatrix{ X' \ar[r]_ k \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ l \ar[d]_{g'} & Y \ar[d]^ g \\ Z' \ar[r]^ m & Z } \]

of ringed spaces. Then the base change maps of Remark 20.28.3 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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