Lemma 76.7.7. Let $S$ be a scheme. Let

be a commutative diagram of algebraic spaces over $S$. Then we have

as maps $(f \circ g)^*\Omega _{X/Y} \to \Omega _{X''/Y''}$.

Lemma 76.7.7. Let $S$ be a scheme. Let

\[ \xymatrix{ X'' \ar[d] \ar[r]_ g & X' \ar[d] \ar[r]_ f & X \ar[d] \\ Y'' \ar[r] & Y' \ar[r] & Y } \]

be a commutative diagram of algebraic spaces over $S$. Then we have

\[ c_{f \circ g} = c_ g \circ g^* c_ f \]

as maps $(f \circ g)^*\Omega _{X/Y} \to \Omega _{X''/Y''}$.

**Proof.**
Omitted. Hint: Use the characterization of $c_ f, c_ g, c_{f \circ g}$ in terms of the effect these maps have on local sections.
$\square$

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