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Exercise Let $U: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $\operatorname{\mathcal{C}}_{/U}$ be the slice category of Construction By virtue of Remark, the objects of $\operatorname{\mathcal{C}}_{/U}$ can be identified with pairs $(Y, \alpha )$, where $Y$ is an object of $\operatorname{\mathcal{C}}$ and $\alpha : \underline{Y} \rightarrow U$ is a natural transformation of functors. Show that $\alpha $ exhibits $Y$ as a limit of $U$ (in the sense of Definition if and only if the pair $(Y, \alpha )$ is a final object of the category $\operatorname{\mathcal{C}}_{/U}$. Similarly, show that a natural transformation $\beta : U \rightarrow \underline{Y}$ exhibits $Y$ as a colimit of $U$ if and only if the pair $(Y,\beta )$ determines an initial object of the coslice category $\operatorname{\mathcal{C}}_{U/}$.