In the paper [Reconstruction of hidden symmetries](https://www.mathematik.uni-muenchen.de/%7Epareigis/Papers/Hidsym.pdf) of Bodo Pareigis in the subsection "3.1 Reconstruction of coalgebras" there is the following proposition (3.3.).


Let $\mathcal{C}$ be a braided monoidal category, $\mathcal{A}$ a $\mathcal{C}$-monoidal category and $\mathcal{B}$ a $\mathcal{C}$-category such as $\omega : \mathcal{B}\longrightarrow\mathcal{A}$ a $\mathcal{C}$-functor. If the functor
\begin{equation\*}
\mathrm{Nat}\_{\mathcal{C}}(\omega,\omega\otimes-) : \mathcal{A}\longrightarrow\mathrm{Set}
\end{equation\*}
is representable by an object $C$ (also denoted as $\mathrm{coend}\_{\mathcal{C}}(\omega)$), then $C$ is a coalgebra in $\mathcal{A}$.


In the proof of this proposition the author just says, that the comultiplication $\Delta$ und counit $\epsilon$ are uniquely definied by $(1\_{\omega}\otimes\Delta)\circ\delta = (\delta\otimes 1\_{C})\circ\delta$ and $(1\_{\omega}\otimes\epsilon)\circ\delta = \rho\_{\omega}^{-1}$, where $\delta : \omega\longrightarrow\omega\otimes C$ is the universal arrow of the representation , but he doesn't prove the coassociativity and the counit axioms. I tried it myself and in fact it is straightforward to show the coassiciativity and that $\epsilon$ is a left counit. For example the diagram
$\require{AMScd}$
\begin{CD}
\omega @>{\delta}>> \omega\otimes C\\
@VV{\delta}V @VV{1\_{\omega}\otimes\Delta}V \\
\omega\otimes C @>{\delta\otimes 1\_{C}}>> \omega\otimes C\otimes C \\
@V{1\_{\omega}\otimes\lambda\_{C}^{-1}}V{\rho\_{\omega}^{-1}\otimes 1\_{C}}V @VV{1\_{\omega}\otimes\epsilon\otimes 1\_{C}}V \\
\omega\otimes 1\otimes C @= \omega\otimes 1\otimes C
\end{CD}
shows that $\epsilon$ is a left counit. However I can not prove that $\epsilon$ is also a right counit, i.g. $\rho\_{C}^{-1} = (1\_{C}\otimes\epsilon)\circ\Delta$. Does anyone know how to do that? Or maybe the author means by a coalgebra something more general, a coalgebra just with a left counit?



Reconstruction of coalgebras
The diagram
$\require{AMScd}$
\begin{CD}
\omega @>{\delta}>> \omega\otimes C\\
@VV{\delta}V @VV{1\_{\omega}\otimes\Delta}V \\
\omega\otimes C @>{\delta\otimes 1\_{C}}>> \omega\otimes C\otimes C \\
@V{1\_{\omega}\otimes\epsilon}VV @VV{1\_{\omega\otimes C}\otimes\epsilon}V \\
\omega\otimes1 @>\delta\otimes1\_1>> \omega\otimes C\otimes1
\end{CD}
commutes and moreover the left vertical composite is $\rho\_\omega^{-1}$ by requirement.


(The lower square commutes since it is the tensor product of two commuting squares,
$$
\begin{CD}
\omega@>\delta>> \omega\otimes C\\
@| @| \\
\omega@>\delta>> \omega\otimes C
\end{CD}\quad\bigotimes\quad
\begin{CD}
C @= C \\
@VV\epsilon V @VV\epsilon V \\
1 @= 1
\end{CD})
$$


