### Equivalence to descending Precosheaf

We start by describing the structure of an $${\overline{{\mathbb{E}}}}$$-category on the the category of set-valued precosheaves on $$\overline{{\mathbb{R}}}_{-\infty}$$. We proceed similar to example 10. To this end let $$\mathcal{U}$$ be the topology or intersection-base of $$\overline{{\mathbb{R}}}_{-\infty}$$ (here they are the same), $$\mathcal{Q}$$ the intersection-base of $$\overline{D}$$, and let $\overline{s} \colon {\overline{{\mathbb{E}}}}\mapsto {\operatorname{End}(\mathcal{U})}, (a, b) \mapsto \begin{cases} (s^b)^{+1} & b > -\infty \\ (s^{-b})^{-1} & b < \infty . \end{cases}$ We note that this definition is not over-determined and that $$\overline{s}$$ is monotone. Now post-composition with the precomposition functor $$\tilde{\mathcal{S}}$$ yields the smoothing functor $$\overline{\mathcal{S}} := \tilde{\mathcal{S}} \circ \overline{s}$$. We note that $$\overline{\mathcal{S}}((a, b)) = s^b_p$$ for $$b > -\infty$$ and $$\overline{\mathcal{S}}((a, b)) = s^{-b}_*$$ for $$b < \infty$$. Now we have the continuous map $$\pi^2$$ from $$\overline{D}$$ to $$\overline{{\mathbb{R}}}_{-\infty}$$ and we aim to show that $$\pi^2_*$$ and $$\pi^2_p$$ are $$1$$-homomorphisms between the corresponding $$D$$-categories and that $$\eta^{\pi^2}$$ is a $$2$$-homomorphism. For $$(a, b) \in D$$ we have $$\pi^2 \circ S^{(a, b)} = s^b \circ \pi^2$$ and with this we convince ourselves that the diagram $\xymatrix@+=3pc{ \mathcal{Q} \ar[d]_{\big(S^{(a, b)}\big)^{+1}} \ar[r]^{(\pi^2)^{+1}} & \mathcal{U} \ar[d]|-{(s^b)^{+1}} \ar[r]^{(\pi^2)^{-1}} & \mathcal{Q} \ar[d]^{\big(S^{(a, b)}\big)^{+1}} \\ \mathcal{Q} \ar[r]_{(\pi^2)^{+1}} & \mathcal{U} \ar[r]_{(\pi^2)^{-1}} & \mathcal{Q} }$ commutes for all $$(a, b) \in D$$. (We note that lemma 5 is the commutativity of the left square.)

• Lemma. The functor $$\pi^2_*$$ is a $$1$$-homomorphism of $$D$$-categories.

• Proof. This follows from the commutativity of the right square in the above diagram and the monotonicity of post-composition by $$(\pi^2)^{-1}$$.

• Lemma. The functor $$\pi^2_p$$ is a $$1$$-homomorphism of $$D$$-categories.

• Proof. This follows from the commutativity of the left square in the above diagram and the monotonicity of post-composition by $$(\pi^2)^{+1}$$.

• Lemma. The natural transformation $$\eta^{\pi^2}$$ is a $$2$$-homomorphism of $$D$$-categories.

• Proof. This follows from the commutativity of the outer square in the above diagram.

The previous three lemmata have the following

• Corollary. The full subcategory of set-valued precosheaves $$F$$ on $$\overline{D}$$, with $$\eta^{\pi^2}_F$$ an isomorphism, is a sub-$$D$$-category in the sense that it is invariant under $$\mathcal{S}(\mathbf{a})$$ for all $$\mathbf{a} \in D$$.

Now the functor $$\pi^2_*$$ is full and faithful on the category of precosheaves $$F$$, with $$\eta^{\pi^2}_F$$ an isomorphism. Since this is also a $$D$$-category and $$\pi^2_*$$ is a $$1$$-homomorphism on this $$D$$-category, $$\pi^2_*$$ yields bijections of interleavings.

Now let $$f \colon X \rightarrow {\mathbb{R}}$$ a continuous function.

1. Lemma. The homomorphism $$(\eta^{\pi^2} \circ \mathcal{C} \circ \mathcal{E})_f$$ from $$\mathcal{C} \mathcal{E} f$$ to $$\pi^2_p \pi^2_* \mathcal{C} \mathcal{E} f$$ is an isomorphism.

• Proof. For $$a < r < b$$ we set $$U := (a, \infty] \times [-\infty, b)$$, then we have $$(\mathcal{C} \mathcal{E} f)(U) = \Lambda({\operatorname{epi}}f \cap X \times (a, b))$$ and $$(\pi^2_p \pi^2_* \mathcal{C} \mathcal{E} f)(U) = \Lambda({\operatorname{epi}}f \cap X \times [-\infty, b))$$. Now let $$i$$ be the inclusion of $${\operatorname{epi}}f \cap X \times [r, b)$$ into $${\operatorname{epi}}f \cap X \times (a, b)$$ and let $$j$$ be the inclusion of $${\operatorname{epi}}f \cap X \times (a, b)$$ into $${\operatorname{epi}}f \cap X \times [-\infty, b)$$. First we prove that $$\Lambda(i)$$ is a bijection. Let $$(x, y) \in {\operatorname{epi}}f \cap X \times (a, r)$$, then $$c \colon [0, 1] \rightarrow {\operatorname{epi}}f \cap X \times (a, b), t \mapsto (x, t(r-y) + y)$$ defines a path from $$(x, y)$$ to $$(x, r)$$, hence $$\Lambda(i)$$ is surjective. Now let $$R \colon {\operatorname{epi}}f \cap X \times (a, b) \rightarrow {\operatorname{epi}}f \cap X \times [r, b), (x, y) \mapsto (x, \max \{r, y\})$$, then $$R \circ i = {\operatorname{id}}$$ and thus $$\Lambda(R) \circ \Lambda(i) = {\operatorname{id}}$$, hence $$\Lambda(i)$$ is injective. By a similar argument we obtain that $$\Lambda(j \circ i) = \Lambda(j) \circ \Lambda(i)$$ is bijective. Now that both $$\Lambda(i)$$ and $$\Lambda(j) \circ \Lambda(i)$$ are bijective, we also have that $$\Lambda(j)$$ is bijective. But $$\Lambda(j)$$ is just $$(\eta^{\pi^2} \circ \mathcal{C} \circ \mathcal{E})_{f ~ U}$$ and this implies the claim.

Now let $$g \colon Y \rightarrow {\mathbb{R}}$$ be another continuous function.

1. Corollary. The interleavings of $$\mathcal{C} \mathcal{E} f$$ and $$\mathcal{C} \mathcal{E} g$$ are in bijection to those of $$\pi^2_* \mathcal{C} \mathcal{E} f$$ and $$\pi^2_* \mathcal{C} \mathcal{E} g$$.