Remark 8.1.0.3 ($\operatorname{Tw}(\operatorname{\mathcal{C}})$ as a Category of Elements). Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ determines a functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , \bullet ): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$. The twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ of Construction 8.1.0.1 can be identified with the category of elements $\int _{ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(\bullet , \bullet )$ (see Construction 5.2.6.1).
It follows that the functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(\bullet , \bullet )$ is determined (up to canonical isomorphism) by the datum of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ together with the forgetful functor $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ of Remark 8.1.0.2 (see Corollary 5.2.7.5).