@@ 65,7 65,7 @@ transformations of the smoothing functor
 To this end let $\mathbf{a}, \mathbf{b} \in -D$ with
 $\mathbf{a} \preceq \mathbf{b}$.
 
-* **Lemma.**
+(@shiftDescJoinTree) **Lemma.**
 We have
 $(\overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ
   \pi^2_* \circ \mathcal{C} \circ \mathcal{E})_f =

          

@@ 49,7 49,7 @@ We refer to $\mathcal{S}$ as the
 
 * *Remark.*
 If $\mathbf{C}$ is a strict $-D$-category with smoothing functor $\mathcal{S}$,
-then the opposite category $\mathcal{C}^{\op}$ is a strict
+then the opposite category $\mathbf{C}^{\op}$ is a strict
 $D$-category with smoothing functor $\mathcal{S}(- (\_))$.
 
 Now we define interleavings in $-D$-categories.

          

@@ 26,3 26,4 @@ 
 [Godement product]: https://ncatlab.org/nlab/show/Godement+product
 [strict 2-category]: https://ncatlab.org/nlab/show/strict+2-category
 [galois]: https://en.wikipedia.org/wiki/Galois_connection#.28Monotone.29_Galois_connection
+[category of arrows]: https://unapologetic.wordpress.com/2007/05/23/arrow-categories/

          

@@ 108,12 108,12 @@ to the category of set-valued precosheav
 such that $\mathcal{C} = \tilde{\mathcal{C}}((\_, \mathbf{o}))$.
 Applying the same procedure to an arbitrary functor $F$
 on the category of $\R$-spaces
-is what we name a *positive persistence-enhancement of $F$*.
+is what we name a *positive persistence-enhancement for $F$*.
 Now let $F$ be a functor from the category of $\R$-spaces
 to some category $\mathbf{C}$
 
 * **Definition** (Positive Persistence-Enhancement)**.**
-A *postive persistence-enhancement of $F$*
+A *postive persistence-enhancement for $F$*
 is the structure of a strict $D$-category on
 $\mathbf{C}$ together with a $1$-homomorphism $\tilde{F}$ from
 $\mathbf{F}$ to $\mathbf{C}$ such that $F = \tilde{F}((\_, \mathbf{o}))$.

          
@@ 131,7 131,7 @@ see [the section on homomorphisms in $D$
 and the previous corollary.
 
 * *Example.*
-To conclude this section we provide a positive persistence-enhancements for
+As a first example we provide a positive persistence-enhancements for
 the Reeb precosheaf $\mathcal{C}$.
 In the previous section we already defined the structure of a
 strict $D$-category on the category of precosheaves.

          
@@ 141,7 141,7 @@ To this end we set
 $\Delta^{\mathbf{a}} \colon \R \rightarrow \Ec,
  t \mapsto (t, t) - \mathbf{a}$ 
 and
-$\overline{C}((f, \mathbf{a})) :=
+$\overline{\mathcal{C}}((f, \mathbf{a})) :=
  (\Delta^{\mathbf{a}} \circ f)_* \Lambda$ for any $\mathbf{a} \in D$.
 Now let
 $\varphi \colon (f, \mathbf{a}) \rightarrow (g, \mathbf{b})$

          
@@ 229,3 229,32 @@ defines a $1$-homomorphism of strict $D$
     $\R$-space and $r \in \R$
     we set
 -->
+
+Now let $F$ and $G$ be functors from the category of $\R$-spaces
+to some $D$-category $\mathbf{C}$ with smoothing functor $\mathcal{S}$
+and let $\eta \colon F \rightarrow G$
+be a natural transformation.
+We consider $\eta$ a functor from the category of $\R$-spaces
+to the [category of arrows][] in $\mathbf{C}$.
+Moreover we consider the [category of arrows][] in $\mathbf{C}$
+a $D$-category with smoothing functor $\mathcal{S}$.
+(We just apply the smoothing functor to the homomorphisms.)
+Then a (positive) persistence-enhancement of
+$\eta$ with smoothing functor $\mathcal{S}$
+is already determined by the corresponding enhancements for $F$ and $G$.
+Now suppose $\tilde{F}$ and $\tilde{G}$ are arbitrary persistence-enhancements
+for $F$ and $G$ both with smoothing functor $\mathcal{S}$.
+
+* **Definition.**
+We say *$\mathcal{S}$, $\tilde{F}$, and $\tilde{G}$
+combine to a persistence-enhancement of $\eta$*
+if the map
+$(f, \mathbf{a}) \mapsto (\mathcal{S}(\mathbf{a}) \circ \eta)_f$
+is a natural transformation from $\tilde{F}$ to $\tilde{G}$.
+
+(@combine2Hom) *Remark.*
+If $\mathcal{S}$, $\tilde{F}$, and $\tilde{G}$
+combine to a persistence-enhancement of $\eta$,
+then
+$(f, \mathbf{a}) \mapsto (\mathcal{S}(\mathbf{a}) \circ \eta)_f$
+is a $2$-homomorphism from $\tilde{F}$ to $\tilde{G}$.

          

@@ 142,7 142,7 @@ and this implies the claim.
 
 Now let $g \colon Y \rightarrow \R$ be another continuous function.
 
-* **Corollary.**
+(@etaCEIsoCor) **Corollary.**
 The interleavings of
 $\mathcal{C} \mathcal{E} f$ and $\mathcal{C} \mathcal{E} g$
 are in bijection to those of

          
@@ 295,7 295,7 @@ the interleavings of
 with respect to the $-D$-category structure given by
 $\overline{\mathcal{S}}$, are in canonical bijection with those
 given by the $\overline{\mathcal{S}}$-induced structure of a $D$-category.
-So in conjunction with the last corollary from the previous subsection
+So in conjunction with corollary @etaCEIsoCor from the previous subsection
 we have the following
 
 (@bijD) **Propostion.**

          

@@ 41,6 41,10 @@ sections:
 - title: Equality of Interleaving Distances
   files:
   - EqualityOfInterlDist.md
+- title: Relation to the Join Precosheaf
+  name: join-precosheaf
+  files:
+  - joinPrecosheaf.md
 - title: Appendix
   sections:
   - title: Constructible Spaces over the Reals