Kerodon

Remark 8.2.4.2. In the situation of Definition 8.2.4.1, if the diagram (8.40) exhibits $\lambda $ as represented by $G$ if and only if the functor $\widetilde{G}$ carries each isomorphism in $\operatorname{\mathcal{C}}_{+}$ (regarded as an object of the $\infty $-category $\operatorname{Tw}(\operatorname{\mathcal{C}}_{+})$) to a universal object of $\operatorname{\mathcal{C}}$. This follows from Remark 8.2.1.2, since every isomorphism in $\operatorname{\mathcal{C}}_{+}$ is isomorphic to an identity morphism (when viewed as an object of $\operatorname{Tw}(\operatorname{\mathcal{C}}_{+})$).