Definition 24.2.1. Let $\mathcal{C}$ be a category. We denote $\text{SR}(\mathcal{C})$ the category of semi-representable objects defined as follows

1. objects are families of objects $\{ U_ i\} _{i \in I}$, and

2. morphisms $\{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ are given by a map $\alpha : I \to J$ and for each $i \in I$ a morphism $f_ i : U_ i \to V_{\alpha (i)}$ of $\mathcal{C}$.

Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be an object of $\mathcal{C}$. The category of semi-representable objects over $X$ is the category $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$.

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