Lemma 101.39.5. Assume given a 2-commutative diagram
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
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
and some choices of y \circ j \to f \circ x.
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