Lemma 4.32.7. Assume $p : \mathcal{S} \to \mathcal{C}$ is a fibred category. Assume given a choice of pullbacks for $p : \mathcal{S} \to \mathcal{C}$.

For any pair of composable morphisms $f : V \to U$, $g : W \to V$ there is a unique isomorphism

\[ \alpha _{g, f} : (f \circ g)^\ast \longrightarrow g^\ast \circ f^\ast \]as functors $\mathcal{S}_ U \to \mathcal{S}_ W$ such that for every $y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ the following diagram commutes

\[ \xymatrix{ g^\ast f^\ast y \ar[r] & f^\ast y \ar[d] \\ (f \circ g)^\ast y \ar[r] \ar[u]^{(\alpha _{g, f})_ y} & y } \]If $f = \text{id}_ U$, then there is a canonical isomorphism $\alpha _ U : \text{id} \to (\text{id}_ U)^*$ as functors $\mathcal{S}_ U \to \mathcal{S}_ U$.

The quadruple $(U \mapsto \mathcal{S}_ U, f \mapsto f^*, \alpha _{g, f}, \alpha _ U)$ defines a pseudo functor from $\mathcal{C}^{opp}$ to the $(2, 1)$-category of categories, see Definition 4.28.5.

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