The Stacks project

Lemma 8.8.3. Notation and assumptions as in Lemma 8.8.2. There is a canonical equivalence of categories

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Fib}/\mathcal{C}}(\mathcal{S}, \mathcal{X}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Stacks}/\mathcal{C}}(\mathcal{S}', \mathcal{X}) \]

given by the constructions in the proof of the aforementioned lemma.

Proof. Omitted. $\square$

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