The Stacks project

Lemma 4.40.3. Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Assume that $\mathcal{X}$, $\mathcal{Y}$ are representable by objects $X$, $Y$ of $\mathcal{C}$. Then

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{X}, \mathcal{Y}) \Big/ 2\text{-isomorphism} = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, Y) \]

More precisely, given $\phi : X \to Y$ there exists a $1$-morphism $f : \mathcal{X} \to \mathcal{Y}$ which induces $\phi $ on isomorphism classes of objects and which is unique up to unique $2$-isomorphism.

Proof. By Example 4.38.7 we have $\mathcal{C}/X = \mathcal{S}_{h_ X}$ and $\mathcal{C}/Y = \mathcal{S}_{h_ Y}$. By Lemma 4.39.6 we have

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{X}, \mathcal{Y}) \Big/ 2\text{-isomorphism} = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ X, h_ Y) \]

By the Yoneda Lemma 4.3.5 we have $\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ X, h_ Y) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, Y)$. $\square$

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