Kerodon

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Definition 7.1.0.1. Let $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. Let $Y$ be an object of $\operatorname{\mathcal{C}}$ and let $\alpha : \underline{Y} \rightarrow U$ be a natural transformation of functors. We say that the natural transformation $\alpha $ exhibits $Y$ as a limit of $F$ if the following condition is satisfied:

$(\ast )$

For every object $X \in \operatorname{\mathcal{C}}$, composition with $\alpha $ induces a bijection from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to the set $\operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{K}},\operatorname{\mathcal{C}})}( \underline{X}, F )$ of natural transformations from $\underline{X}$ to $U$.