Lemma 4.35.17. Let $\mathcal{C}$ be a category. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. Assume we have a $2$-commutative diagram
\[ \xymatrix{ \mathcal{X}' \ar[rd]_ f & \mathcal{X} \ar[l]^ a \ar[d]^ F \ar[r]_ b & \mathcal{X}'' \ar[ld]^ g \\ & \mathcal{Y} } \]
where $a$ and $b$ are equivalences of categories over $\mathcal{C}$ and $f$ and $g$ are categories fibred in groupoids. Then there exists an equivalence $h : \mathcal{X}'' \to \mathcal{X}'$ of categories over $\mathcal{Y}$ such that $h \circ b$ is $2$-isomorphic to $a$ as $1$-morphisms of categories over $\mathcal{C}$. If the diagram above actually commutes, then we can arrange it so that $h \circ b$ is $2$-isomorphic to $a$ as $1$-morphisms of categories over $\mathcal{Y}$.
Proof.
We will show that both $\mathcal{X}'$ and $\mathcal{X}''$ over $\mathcal{Y}$ are equivalent to the category fibred in groupoids $\mathcal{X} \times _{F, \mathcal{Y}, \text{id}} \mathcal{Y}$ over $\mathcal{Y}$, see proof of Lemma 4.35.16. Choose a quasi-inverse $b^{-1} : \mathcal{X}'' \to \mathcal{X}$ in the $2$-category of categories over $\mathcal{C}$. Since the right triangle of the diagram is $2$-commutative we see that
\[ \xymatrix{ \mathcal{X} \ar[d]_ F & \mathcal{X}'' \ar[l]^{b^{-1}} \ar[d]^ g \\ \mathcal{Y} & \mathcal{Y} \ar[l] } \]
is $2$-commutative. Hence we obtain a $1$-morphism $c : \mathcal{X}'' \to \mathcal{X} \times _{F, \mathcal{Y}, \text{id}} \mathcal{Y}$ by the universal property of the $2$-fibre product. Moreover $c$ is a morphism of categories over $\mathcal{Y}$ (!) and an equivalence (by the assumption that $b$ is an equivalence, see Lemma 4.31.7). Hence $c$ is an equivalence in the $2$-category of categories fibred in groupoids over $\mathcal{Y}$ by Lemma 4.35.9.
We still have to construct a $2$-isomorphism between $c \circ b$ and the functor $d : \mathcal{X} \to \mathcal{X} \times _{F, \mathcal{Y}, \text{id}} \mathcal{Y}$, $x \mapsto (p(x), x, F(x), \text{id}_{F(x)})$ constructed in the proof of Lemma 4.35.16. Let $\alpha : F \to g \circ b$ and $\beta : b^{-1} \circ b \to \text{id}$ be $2$-isomorphisms between $1$-morphisms of categories over $\mathcal{C}$. Note that $c \circ b$ is given by the rule
\[ x \mapsto (p(x), b^{-1}(b(x)), g(b(x)), \alpha _ x \circ F(\beta _ x)) \]
on objects. Then we see that
\[ (\beta _ x, \alpha _ x) : (p(x), x, F(x), \text{id}_{F(x)}) \longrightarrow (p(x), b^{-1}(b(x)), g(b(x)), \alpha _ x \circ F(\beta _ x)) \]
is a functorial isomorphism which gives our $2$-morphism $d \to b \circ c$. Finally, if the diagram commutes then $\alpha _ x$ is the identity for all $x$ and we see that this $2$-morphism is a $2$-morphism in the $2$-category of categories over $\mathcal{Y}$.
$\square$
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