I've read about monoidal categories, and I understand them, however I'm still confused about calling the associator, left unitor, and right unitor natural isomorphisms.
Natural isomorphisms are just natural transformations, where every component has an inverse in the category itself.
But I wonder what kind of natural transformation they are? What are their domain and codomain? E.g. the left unitor:
$$ \lambda: (I ~\otimes ~ -) \cong - $$
A natural transformation has a component for every object in the domain. I.e. it should also have a component for $(c,c)$ for any object $c$. But what would $\lambda(c ~\otimes ~ c)$ be? just an identity function?
The same applies to the associator, what e.g. would $\alpha(I \otimes I)$ be?