Lemma 62.6.3. Let $f : X \to Y$ and $g : Y \to Z$ be separated and locally quasi-finite morphisms. There is a canonical isomorphism $(g \circ f)^! \to f^! \circ g^!$. Given a third locally quasi-finite morphism $h : Z \to T$ the diagram

$\xymatrix{ (h \circ g \circ f)^! \ar[r] \ar[d] & f^! \circ (h \circ g)^! \ar[d] \\ (g \circ f)^! \circ h^! \ar[r] & f^! \circ g^! \circ h^! }$

commutes.

Proof. By uniqueness of adjoint functors, this immediately translates into the corresponding (dual) statement for the functors $f_!$. See Lemma 62.4.12. $\square$

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