I am trying to prove Proposition 7.2 from this paper:
$\textbf{Proposition 7.2:}$ Let $A$ be a finite-dimensional quasi-Hopf algebra over a field $\mathbb{K}$. The end object $\Gamma$ in $\textbf{Rep A}$ can be chosen to be the adjoint representation $\Gamma=(A, \rho_{A}^{adj})$, together with the dinatural family $j=(j_{M}: A \rightarrow M \otimes M^{*})$ given by: $$j_{M}: a \mapsto \sum \limits_{i}(a \centerdot m_{i}) \otimes m_{i}^{*},$$ where $m_{i}$ is a basis in $M$ and $m_{i}^{*}$ is the dual basis.
I am interested in this result for $A$ being a finite-dimensional ribbon Hopf algebra. Let me recall some definitions to make my question more clear. The (left) adjoint action $\rho_{A}^{adj}: A \otimes A \rightarrow A$ is given by $a \centerdot h = \Sigma a_{(1)}hS(a_{(2)})$.
Let $\mathcal{C}$ be a braided monoidal category with right duality. $A$ is a ribbon Hopf algebra, $Rep A$ is the category of finite-dimensional representations of $A$. The morphisms are intertwiners.
$\textbf{Definition:}$ A dinatural transformation in $\mathcal{C}$ from an object $B \in Ob(\mathcal{C})$ is a family of morphisms $d = \{ d_{X}: B \rightarrow X \otimes X^{*} \} _{X \in Ob(\mathcal{C})}$ in $\mathcal{C}$ such that the following diagram commutes for all $X, Y \in Ob(\mathcal{C})$ and morphism $f: X \rightarrow Y$:
An end of $\mathcal{C}$ is a pair $(E,j)$, where $E \in Ob(\mathcal{C})$, and $j = \{ j_{X}: E \rightarrow X \otimes X^{*} \}_{X \in \mathcal{C}}$ is a dinatural tranformation which is universal: for any dinatural transformation $d = \{ d_{X}: D \rightarrow X \otimes X^{*} \}_{X \in Ob(\mathcal{C})}$ there is a unique morphism $\varphi: D \rightarrow E$, such that for any $X,Y \in Ob(\mathcal{C})$ the triangles in the the diagram commute:
This definition is a simplified version of the definition of an end of a functor, which we find in the classical books such as Mac Lane. In our case the functor is $F: \mathcal{C} \times \mathcal{C}^{op} \rightarrow \mathcal{C}.$
Back to my question: the proof of the Proposition is divided into the following steps: (1) $j_{M}$ is an intertwiner; (2) $j_{M}$ is a dinatural transformation; (3) there is an intertwiner $\varphi: D \rightarrow A$ for $D \in Ob(\mathcal{C})$; (4) $\varphi$ makes the triangles in the definition of an end commute; (5) $\varphi$ is unique.
I managed to prove steps (1) and (2) (i can include them here, if you wish) and I am stuck at step (3). I try to decompose the map $\varphi: D \rightarrow A$ as follows: $$\varphi: D \xrightarrow{d_{A}} A \otimes A^{*} \xrightarrow{} A \otimes A \xrightarrow{\rho_{adj}} A, $$ where $d_{A}$ is the dinatural transformation from $D$. My question is now what the map $A \otimes A^{*} \rightarrow A \otimes A$ could be?
Sorry for the long question, but I wanted to include the definitions to make it more clear. Any help or comment is highly appreciated! Thank you very much in advance.
$\textbf{UPDATE:}$ My first attempt for the composition $\varphi$ was wrong, so I changed it to the correct one.