Suppose you have a monoidal $\mathbb{C}$-linear category $\mathcal{C}$ It is stated in nlab (See here https://ncatlab.org/nlab/show/semisimple+category#:~:text=Definition,of%20finitely%20many%20simple%20objects.) that if your category is idempotent complete (i.e. every idempotent splits) and there are objects $X_i$ such that $Hom(X_i,X_j)=\delta_{i,j}$ with the following map being an isomorphism for any objects $V,W$: $$\varphi_{V,W}:\oplus Hom(V,X_i)\otimes Hom(X_i,W)\rightarrow Hom(V,W)$$ Where this is taking the direct sum of tensors to the sum of the composition. Then one can write any object in the category as a direct sum of these objects $X_i$. In this link they seem to try and show that $Hom(V,X_i)$ is the monoidal dual of $Hom(X_i,V)$ in the category $Vec$ and hence it is isomorphic to the actual dual space (in terms of linear functionals). It defines some evaluation and coevaluation maps and then claims that the snake equation holds but I think this is not clear. I'd suppose what is meant is that you can define evaluation and coevaluation maps by $Hom(X_i,V)\otimes Hom(V,X_i)\rightarrow \mathbb{C}$ is just the composition (note $Hom(X_i,X_i)\cong \mathbb{C}$) and also considering the inverse of $id_V$ under the map $\varphi_{V,V}$ and cutting down to the projection in the $i^{th}$ summand but its not clear to me that these are actually evaluation and coevaluation maps (i.e. the snake equations hold).
To be more precise I have a bit of a problem with how to deal with the map $\varphi_{V,V}$ being an isomorphism. As I've stated since the map is an isomorphism there are $f_i^l\in Hom(V,X_i)$ and $g_i^l\in Hom(X_i,V)$ such that $\sum_{i,l}g_i^lf_i^l=id_V$ taking the inverse and then cutting down the projection just gives the $\sum_l g_i^l\otimes f_i^l$ as the coevaulation map evaluated at $1$. Now trying to do the snake equations (e.g. $(id\otimes ev)(coev\otimes id)$ on $h\in Hom(X_i,V)$) and you get $\sum_l g_i^l(f_i^lh)$ how do you know this is $h$ again i.e. that these $g_i^l,f_i^l$ act as an inner product basis?