Lemma 4.36.3. Let $\mathcal{C}$ be a category. Let $\mathcal{S}$ be a fibred category over $\mathcal{C}$. Then $\mathcal{S}$ is split if and only if for some choice of pullbacks (see Definition 4.33.6) the pullback functors $(f \circ g)^*$ and $g^* \circ f^*$ are equal.

**Proof.**
This is immediate from the definitions.
$\square$

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