Below $\mathcal{I}\to \textbf{Set}$ is a functor(diagram) between two categories while $M_i$ are all sets. There are no restrictions on $\mathcal{I}$. I cannot see why the equivalence of elements $m_i\in M_i$ and $m_{i'}\in M_{i'}$, $m_i\sim m_{i'}$ are related to the chain of morphisms below. Isn't it just an arrow $i\to i'$? What are the $i_0,...i_{2n}$ doing here?
The image below is from here.
