Lemma 101.39.5. Assume given a $2$-commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[dd]_ j & \mathcal{X} \ar[d]^ f \\ & \mathcal{Y} \ar[d]^ g \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-z & \mathcal{Z} }$

Choose a $2$-arrow $\gamma : z \circ j \to g \circ f \circ x$. Let $\mathcal{C}$ be the category of dotted arrows for the outer rectangle and $\gamma$. Let $\mathcal{C}'$ be the category of dotted arrows for the square

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-{f \circ x} \ar[d]_ j & \mathcal{Y} \ar[d]^ g \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-z & \mathcal{Z} }$

and $\gamma$. Then $\mathcal{C}$ is equivalent to a category $\mathcal{C}''$ which has the following property: there is a functor $\mathcal{C}'' \to \mathcal{C}'$ which turns $\mathcal{C}''$ into a category fibred in groupoids over $\mathcal{C}'$ and whose fibre categories are categories of dotted arrows for certain squares of the form

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[d]_ j & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-y & \mathcal{Y} }$

and some choices of $y \circ j \to f \circ x$.

Proof. This lemma is a special case of Categories, Lemma 4.44.3. $\square$

There are also:

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