The following definition^{[4]} is from (Morozov, Beketayev, and Weber
2013).

**Definition.**Let \(f \colon X \rightarrow {\mathbb{R}}\) be a continuous map we define it’s*join tree*to be the continuous map \((\rho \circ \iota)_f\) from \((\pi \circ \iota)_f ({\operatorname{epi}}f)\) to \({\mathbb{R}}\).

With this definition \(\rho \circ \iota\) is an endofunctor on topological spaces over \({\mathbb{R}}\). Given a continuous map \(f \colon X \rightarrow {\mathbb{R}}\) we have \((\pi \circ \iota)_f \circ \kappa_f = (\rho \circ \kappa)_f \circ \pi_f\), so in somewhat sloppy notation \((\pi \circ \iota) \circ \kappa = (\rho \circ \kappa) \circ \pi\) is a natural transformation from \({\operatorname{id}}\) to \(\rho \circ \iota\). Similarly we have the function \((\gamma \circ \lambda \circ \iota)_f\) defined on the display space \({\operatorname{dis}}(\lambda \circ \iota)_f\) of \((\lambda \circ \iota)_f\). And just as with \(\rho\) we have the natural transformation \((\eta \circ \iota) \circ \kappa = (\gamma \circ \lambda \circ \kappa) \circ \eta\) from \({\operatorname{id}}\) to \(\gamma \circ \lambda \circ \iota\). The two constructions are related via the commutative diagram \[ \xymatrix@C+1pc@R-1pc{ & \rho_f \ar[r]^{(\rho \circ \kappa)_f} \ar[dd]|-{\phi_f} & (\rho \circ \iota)_f \ar[dd]^{(\phi \circ \iota)_f} \\ f \ar[ur]^{\pi_f} \ar[dr]_{\eta_f} \\ & (\gamma \circ \lambda)_f \ar[r]_{(\gamma \circ \lambda \circ \kappa)_f} & (\gamma \circ \lambda \circ \iota)_f } \] given a function \(f \colon X \rightarrow {\mathbb{R}}\). In the section on the Reeb space we considered the left triangle which suggests to replace the Reeb graph functor \(\rho\) and the natural transformation \(\pi\) by \(\gamma \circ \lambda\) and \(\eta\) respectively. Now \(\rho \circ \kappa\) yields a nice and classic map from any Reeb graph to the corresponding join tree, so our replacement of the Reeb graph functor \(\rho\) by \(\gamma \circ \lambda\) is only complete, if also we can replace the join tree functor \(\rho \circ \iota\) and the natural transformation \(\rho \circ \kappa\) and if we can extend \(\phi\) to a natural transformation from \(\rho \circ \kappa\) to it’s replacement. And here the commutative square on the right hand side, suggests we may take \(\lambda \circ \gamma \circ \iota\) as a replacement for the join tree functor \(\rho \circ \iota\) and to take \(\gamma \circ \lambda \circ \kappa\) as a replacement for \(\rho \circ \kappa\), since then we can extend \(\phi\) by \(\phi \circ \iota\) to a natural transformation from \(\rho \circ \kappa\) to \(\gamma \circ \lambda \circ \kappa\). We further note that by corollary 2 in that section the natural transformation \((\phi, \phi \circ \iota)\) from \(\rho \circ \kappa\) to \(\gamma \circ \lambda \circ \kappa\) is isomorphic to the natural transformation \((\eta \circ \rho, \eta \circ \rho \circ \iota)\) from \(\rho \circ \kappa\) to \(\gamma \circ \lambda \circ \rho \circ \kappa\), so our choice of replacements is the same as if we applied \(\gamma \circ \lambda\) to the upper row in the diagram. And by proposition 8 we have a natural isomorphism from \((\gamma \circ \lambda \circ \iota)\) to \(\gamma {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda\) that commutes with \(\gamma \circ \lambda \circ \kappa\) and \(\gamma \circ \eta' \circ \lambda\) sothat we can use the following

**Proposition.**\({\operatorname{id}}_* \lambda\) and \(\gamma {\operatorname{id}}^{-1}\) form a pair of adjoint functors \({\operatorname{id}}_* \lambda \dashv \gamma {\operatorname{id}}^{-1}\) with unit \((\gamma \circ \eta' \circ \lambda) \circ \eta\) and whose counit is an isomorphism.*Proof.*The first statement follows from theorem 1, proposition 3 and the general statement that the two pairs of adjoint functors, when composed in the same way as in our claim, form again a pair of adjoint functors with the unit described as in the claim, see for example https://en.wikipedia.org/wiki/Adjoint_functors#Composition. And for the counit of this composed adjunction we have the formula \({\operatorname{id}}_* \varepsilon {\operatorname{id}}^{-1} \circ \varepsilon'\). By theorem 1 \(\varepsilon\) is an isomorphism, hence \({\operatorname{id}}_* \varepsilon {\operatorname{id}}^{-1}\) is an isomorphism and by proposition 3 \(\varepsilon'\) is an isomorphism and thus our claim follows.**Corollary.**The category of cosheaves on \(\bar{{\mathbb{R}}}\) is equivalent to the reflective subcategory of ascending cosheaf spaces over \({\mathbb{R}}\).*Proof.*By Gabriel and Zisman (1967 Proposition 1.3 or http://ncatlab.org/nlab/show/reflective+subcategory#characterizations) the category of cosheaves on \(\bar{{\mathbb{R}}}\) is equivalent to the reflective subcategory of those spaces \(f \colon X \rightarrow {\mathbb{R}}\) over \({\mathbb{R}}\) for which \((\gamma \circ \eta' \circ \lambda)_f \circ \eta_f\) is an isomorphism. Now suppose this is the case for \(f\), then \(f\) is isomorphic to \(\gamma {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda\) which is in the image of \(\gamma\) and thus a cosheaf space, hence \(\eta_f\) is an isomorphism. From this it follows that \((\gamma \circ \eta' \circ \lambda)_f\) is an isomorphism as well, hence by proposition 8 \((\gamma \circ \lambda \circ\kappa)_f\) is an isomorphism. Now we consider the commutative diagram \[ \xymatrix@C+1pc{ f \ar[r]^{\kappa_f} \ar[d]_{\eta_f} & \iota_f \ar[d]^{(\eta \circ \iota)_f} \\ (\gamma \circ \lambda)_f \ar[r]_{(\gamma \circ \lambda \circ \kappa)_f} & (\gamma \circ \lambda \circ \iota)_f . } \] Hence we have the retract^{[5]}\(R := \eta_f^{-1} \circ (\gamma \circ \lambda \circ \kappa)_f^{-1} \circ (\eta \circ \iota)_f\) from \(\iota_f\) to \(f\). By lemma 5 \(\iota_f\) is ascending, so given \(r \in {\mathbb{R}}\) there is a map \(H_r\) as in definition 4. Now let \(\tilde{H}_r \colon X \times [0, 1] \rightarrow X\) be defined by \(\tilde{H}_r (x, t) = r(H_r(\kappa_f(x), t))\) then \(\tilde{H}_r\) inherits the properties needed in order for \(f\) to be ascending. Conversely if \(f\) is an ascending cosheaf space over \({\mathbb{R}}\), then \(\eta_f\) is an isomorphism since \(f\) is a cosheaf space. And by lemma 6 \(\lambda_f\) is ascending, hence \((\eta' \circ \lambda)_f\) is an isomorphism.

In conclusion \((\gamma \circ \lambda \circ \iota)_f\) is an ascending cosheaf space over \({\mathbb{R}}\) given a function \(f\). It’s cosheaf of connected components \((\lambda \circ \gamma \circ \lambda \circ \iota)_f\) is isomorphic to \((\lambda \circ \iota)_f\) by theorem 1. By lemma 5 and lemma 6 \((\lambda \circ \iota)_f\) is ascending, and thus we have an associated cosheaf \({\operatorname{id}}_* (\lambda \circ \iota)_f\) on \(\bar{{\mathbb{R}}}\) via the adjunction \({\operatorname{id}}_* \dashv {\operatorname{id}}^{-1}\) by proposition 3. By lemma 7 this cosheaf is isomorphic to \({\operatorname{id}}_* \lambda_f\) which is the cosheaf on \(\bar{{\mathbb{R}}}\) associated to \(\gamma {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda_f\) via the adjunction \({\operatorname{id}}_* \lambda \dashv \gamma {\operatorname{id}}^{-1}\). Now applying \({\operatorname{id}}^{-1}\) to \({\operatorname{id}}_* \lambda_f \cong {\operatorname{id}}_* (\lambda \circ \iota)_f\) recovers \((\lambda \circ \iota)_f\), hence \((\gamma \circ \lambda \circ \iota)_f\) and \(\gamma {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda_f\) are isomorphic and thus a posteriori \({\operatorname{id}}_* \lambda_f\) is the cosheaf on \(\bar{{\mathbb{R}}}\) associated to the ascending cosheaf space \((\gamma \circ \lambda \circ \iota)_f\) via the adjunction \({\operatorname{id}}_* \lambda \dashv \gamma {\operatorname{id}}^{-1}\). (Here the author allowed himself some redundance repeating the proof of proposition 8.)