Definition 89.17.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $f: A' \to A$ be a map in $\mathcal{C}_\Lambda$. Let $x \in \mathcal{F}(A)$. The category $\textit{Lift}(x, f)$ of lifts of $x$ along $f$ is the category with the following objects and morphisms.

1. Objects: A lift of $x$ along $f$ is a morphism $x' \to x$ lying over $f$.

2. Morphisms: A morphism of lifts from $a_1 : x'_1 \to x$ to $a_2 : x'_2 \to x$ is a morphism $b : x'_1 \to x'_2$ in $\mathcal{F}(A')$ such that $a_2 = a_1 \circ b$.

The set $\text{Lift}(x, f)$ of lifts of $x$ along $f$ is the set of isomorphism classes of $\textit{Lift}(x, f)$.

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