The Stacks project

Remark 63.3.14. The isomorphisms between functors constructed above satisfy the following two properties:

  1. Let $f : X \to Y$, $g : Y \to Z$, and $h : Z \to T$ be composable morphisms of schemes which are separated and locally of finite type. Then the diagram

    \[ \xymatrix{ (h \circ g \circ f)_! \ar[r] \ar[d] & (h \circ g)_! \circ f_! \ar[d] \\ h_! \circ (g \circ f)_! \ar[r] & h_! \circ g_! \circ f_! } \]

    commutes where the arrows are those of Lemma 63.3.13.

  2. Suppose that we have a diagram of schemes

    \[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_ c & X \ar[d]^ f \\ Y' \ar[d]_{g'} \ar[r]_ b & Y \ar[d]^ g \\ Z' \ar[r]^ a & Z } \]

    with both squares cartesian and $f$ and $g$ separated and locally of finite type. Then the diagram

    \[ \xymatrix{ a^{-1} \circ (g \circ f)_! \ar[d] \ar[rr] & & (g' \circ f')_! \circ c^{-1} \ar[d] \\ a^{-1} \circ g_! \circ f_! \ar[r] & g'_! \circ b^{-1} \circ f_! \ar[r] & g'_! \circ f'_! \circ c^{-1} } \]

    commutes where the horizontal arrows are those of Lemma 63.3.12 the arrows are those of Lemma 63.3.13.

Part (1) holds true because we have a similar commutative diagram for pushforwards. Part (2) holds by the very general compatibility of base change maps for pushforwards (Sites, Remark 7.45.3) and the fact that the isomorphisms in Lemmas 63.3.12 and 63.3.13 are constructed using the corresponding maps of pushforwards.

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