Proof.
By definition, the space \(\mathbb T_2^{\mathcal V_{/C}}(f)\) is the fiber of the functor \(\underline{\mathrm{Alg}}_{\mathbb T_2}(\mathcal V_{/C}) \rightarrow\underline{\mathrm{Alg}}_{[1] \otimes \mathbb E_0}(\mathcal V_{/D})\) at \(f\). By the universal property of the symmetric monoidal structure on the over-category \(\mathcal V_{/C}\), this is equivalent to the functor \(\underline{\mathrm{Alg}}_{\mathbb T_2}(\mathcal V)_{/C} \rightarrow\underline{\mathrm{Alg}}_{[1] \otimes \mathbb E_0}(\mathcal V)_{/C}\). Unwinding the definition of morphisms in over-categories, this results in the desired equivalence. ◻