Evaluation under tensor products Could anyone explain to me the highlighted step in this

calculation? Can I just exaluate the tenor product component wise?
The first step is basically $$(ev_{V}\otimes id_{V})\circ(id_{V}(v)\otimes coev_{V}(v))=(ev_{V}\otimes id_{V})(v^ie_{i}\otimes(\epsilon^{j}\otimes e_{j}))$$ and then the second equality would look like
$$(ev_{V}\otimes id_{V})(v^ie_{i}\otimes(\epsilon^{j}\otimes e_{j}))=(ev_{V}((v^ie_{i}\otimes(\epsilon^{j}\otimes e_{j})))\otimes id_{V}((v^ie_{i}\otimes(\epsilon^{j}\otimes e_{j}))))$$. But I think I'm wrong there.

Yes, you can calculate the tensor product "component-wise", i.e. if $f,g$ are maps, then $(f \otimes g)(x \otimes y) = f(x) \otimes g(y)$. In your case we get that
\begin{align}
(\text{ev}_V \otimes \text{id}_V)(v^ie_i \otimes \epsilon^j \otimes e_j) &= (\text{ev}_V \otimes \text{id}_V)\left((v^ie_i \otimes \epsilon^j) \otimes e_j\right)\\
&= \text{ev}_V(v^ie_i \otimes \epsilon^j) \otimes \text{id}_V(e_j)\\
&= \epsilon^j(v^ie_i) \otimes e_j\\
&= v^i\epsilon^j(e_i) \otimes e_j = v^i \delta_i^j \otimes e_j = v^ie_i = v.
\end{align}
I hope this is helpful!
