Remark 90.17.4 (06JH)—The Stacks project

Remark 90.17.4. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda $ satisfying (RS). Let

\[ \xymatrix{ A_1 \times _ A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A } \]

be a fibre square in $\mathcal{C}_\Lambda $ such that either $A_1 \to A$ or $A_2 \to A$ is surjective. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$. Given lifts $x_1 \to x$ and $x_2 \to x$ of $x$ to $A_1$ and $A_2$, we get by (RS) a lift $x_1 \times _ x x_2 \to x$ of $x$ to $A_1 \times _ A A_2$. Conversely, by Lemma 90.16.2 any lift of $x$ to $A_1 \times _ A A_2$ is of this form. Hence a bijection

\[ \text{Lift}(x, A_1) \times \text{Lift}(x, A_2) \longrightarrow \text{Lift}(x, A_1 \times _ A A_2). \]

Similarly, if $x_1 \to x$ is a fixed lifting of $x$ to $A_1$, then there is a bijection

\[ \text{Lift}(x_1, A_1 \times _ A A_2) \longrightarrow \text{Lift}(x, A_2). \]

Now let

\[ \xymatrix{ A_1' \times _ A A_2 \ar[r] \ar[d] & A_1 \times _ A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1' \ar[r] & A_1 \ar[r] & A } \]

be a composition of fibre squares in $\mathcal{C}_\Lambda $ with both $A'_1 \to A_1$ and $A_1 \to A$ surjective. Let $x_1 \to x$ be a morphism lying over $A_1 \to A$. Then by the above we have bijections

\begin{align*} \text{Lift}(x_1, A_1' \times _ A A_2) & = \text{Lift}(x_1, A_1') \times \text{Lift}(x_1, A_1 \times _ A A_2) \\ & = \text{Lift}(x_1, A_1') \times \text{Lift}(x, A_2). \end{align*}

The Stacks project

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