I have been reading Kassel's Quantum groups and there is something I can not understand.

In Section 4 of chapter $XX$, he introduces the notion of a *Infinitesimal symmetric category*, that is
a strict tensor category $(\mathcal{S}, \bigotimes, I)$ in which all the Hom sets are vector spaces together with, for every object $V,W$ of $\mathcal{S}$, a symmetry $\sigma_{V,W}: V \otimes W \rightarrow W \otimes V $ and an infinitesimal braiding $t_{V,W} :V \otimes W \rightarrow V \otimes W$ having the following properties:

$$\sigma_{V,W} \circ t_{V,W} = t_{W,V} \circ \sigma_{V,W}$$

and

$$ t_{U,V\otimes W} = t_{U,V}\otimes \mathrm{id}_W + (\sigma_{U,V}\otimes \mathrm{id}_W)^{-1} \circ(\mathrm{id}_v \otimes t_{U,W}) \circ (\sigma_{U,V}\otimes \mathrm{id}_W).$$

He then claims that if the infinitesimal symmetric category category $\mathcal{S}$ also has a left duality $V \mapsto V^*$ with structure maps $b_V^0 : I \rightarrow V \otimes V^*$ and $d_V^0 : V^* \otimes V \rightarrow I$, then we have have $$t_{V,W} = \frac12 (C_{V\otimes W}- C_V \otimes \mathrm{id}_W- \mathrm{id}_V \otimes C_W)$$ where the $(C_V : V \rightarrow V)_V$ is a natural family of endomorphisms of $\mathcal{S}$ given by : $$ C_v = - \big(\mathrm{id}_V \otimes(d_v^0 \circ t_{V^*.V})\big) \circ (b_V^0 \otimes \mathrm{id}_V).$$

I simply do not see at all why this is true, so I would glad welcome so kind of explanation. I have tried something using the fact that $(\mathrm{id}_V \otimes d_V^0) \circ (b_V^0 \otimes \mathrm{id}_V) = \mathrm{id}_V$ but this has not been very successful.