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Definition 4.32.1. Let $\mathcal{C}$ be a category. The $2$-category of categories over $\mathcal{C}$ is the $2$-category defined as follows:

  1. Its objects will be functors $p : \mathcal{S} \to \mathcal{C}$.

  2. Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$.

  3. Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_ x) = \text{id}_{p(x)}$ for all $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$.

In this situation we will denote

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}, \mathcal{S}') \]

the category of $1$-morphisms between $(\mathcal{S}, p)$ and $(\mathcal{S}', p')$

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