Remark 105.6.1 (Category of lifts). Consider a diagram

where $\mathcal{X} \subset \mathcal{X}'$ is a thickening of algebraic stacks, $W$ is an algebraic space, and $W \to \mathcal{X}$ is smooth. We will construct a category $\mathcal{C}$ and a functor

(see Properties of Spaces, Definition 65.18.2 for notation) as follows. An object of $\mathcal{C}$ will be a system $(U, U', a, i, x', \alpha )$ which forms a commutative diagram

with commutativity witnessed by the $2$-morphism $\alpha : x \circ a \to x' \circ i$ such that $U$ and $U'$ are algebraic spaces, $a : U \to W$ is étale, $x' : U' \to \mathcal{X}'$ is smooth, and such that $U = \mathcal{X} \times _{\mathcal{X}'} U'$. In particular $U \subset U'$ is a thickening. A morphism

is given by $(f, f', \gamma )$ where $f : U \to V$ is a morphism over $W$, $f' : U' \to V'$ is a morphism whose restriction to $U$ gives $f$, and $\gamma : x' \circ f' \to y'$ is a $2$-morphism witnessing the commutativity in right triangle of the diagram below

Finally, we require that $\gamma $ is compatible with $\alpha $ and $\beta $: in the calculus of $2$-categories of Categories, Sections 4.28 and 4.29 this reads

(more succinctly: $\beta = j^*\gamma \circ f^*\alpha $). Another formulation is that objects are commutative diagrams (105.6.1.1) with some additional properties and morphisms are commutative diagrams (105.6.1.2) in the category $\textit{Spaces}/\mathcal{X}'$ introduced in Properties of Stacks, Remark 99.3.7. This makes it clear that $\mathcal{C}$ is a category and that the rule $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ sending $(U, U', a, i, x', \alpha )$ to $a : U \to W$ is a functor.

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