Remark 62.8.4. Consider a commutative diagram

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

of schemes whose vertical arrows are proper and whose horizontal arrows are separated and locally quasi-finite. Let us label the squares of the diagram $A$, $B$, $C$, $D$ as follows

$\begin{matrix} A & B \\ C & D \end{matrix}$

Then the maps of Lemma 62.8.1 for the squares are (where we use $Rf_* = f_*$, etc)

$\begin{matrix} \gamma _ A : l'_! \circ f''_* \to f'_* \circ k'_! & \gamma _ B : l_! \circ f'_* \to f_* \circ k_! \\ \gamma _ C : m'_! \circ g''_* \to g'_* \circ l'_! & \gamma _ D : m_! \circ g'_* \to g_* \circ l_! \end{matrix}$

For the $2 \times 1$ and $1 \times 2$ rectangles we have four further maps

$\begin{matrix} \gamma _{A + B} : (l \circ l')_! \circ f''_* \to f_* \circ (k \circ k')_* \\ \gamma _{C + D} : (m \circ m')_! \circ g''_* \to g_* \circ (l \circ l')_! \\ \gamma _{A + C} : m'_! \circ (g'' \circ f'')_* \to (g' \circ f')_* \circ k'_! \\ \gamma _{B + D} : m_! \circ (g' \circ f')_* \to (g \circ f)_* \circ k_! \end{matrix}$

By Lemma 62.8.3 we have

$\gamma _{A + B} = \gamma _ B \circ \gamma _ A, \quad \gamma _{C + D} = \gamma _ D \circ \gamma _ C$

and by Lemma 62.8.2 we have

$\gamma _{A + C} = \gamma _ A \circ \gamma _ C, \quad \gamma _{B + D} = \gamma _ B \circ \gamma _ D$

Here it would be more correct to write $\gamma _{A + B} = (\gamma _ B \star \text{id}_{k'_!}) \circ (\text{id}_{l_!} \star \gamma _ A)$ with notation as in Categories, Section 4.28 and similarly for the others. Having said all of this we find (a priori) two transformations

$m_! \circ m'_! \circ g''_* \circ f''_* \longrightarrow g_* \circ f_* \circ k_! \circ k'_!$

namely

$\gamma _ B \circ \gamma _ D \circ \gamma _ A \circ \gamma _ C = \gamma _{B + D} \circ \gamma _{A + C}$

and

$\gamma _ B \circ \gamma _ A \circ \gamma _ D \circ \gamma _ C = \gamma _{A + B} \circ \gamma _{C + D}$

The point of this remark is to point out that these transformations are equal. Namely, to see this it suffices to show that

$\xymatrix{ m_! \circ g'_* \circ l'_! \circ f''_* \ar[r]_{\gamma _ D} \ar[d]_{\gamma _ A} & g_* \circ l_! \circ l'_! \circ f''_* \ar[d]^{\gamma _ A} \\ m_! \circ g'_* \circ f'_* \circ k'_! \ar[r]^{\gamma _ D} & g_* \circ l_! \circ f'_* \circ k'_! }$

commutes. This is true because the squares $A$ and $D$ meet in only one point, more precisely by Categories, Lemma 4.28.2 or more simply the discussion preceding Categories, Definition 4.28.1.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).