What functors are the natural isomorphisms relating in monoidal categories?

Question

I'm having a little difficulty understanding the definition of monoidal category. I intuitively understand what the axioms are expressing, but I'm having a bit of difficulty knowing which functors the natural isomorphisms are actually relating. My weak understanding of the situation is that $\alpha$ is a natural isomorphism between the functor $F: \mathcal{C}\times\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ given by $\otimes(1\times\otimes)$ and the functor $G: \mathcal{C}\times\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ given by $\otimes(\otimes\times 1)$. Likewise, $\lambda$ is a natural isomorphism between $H: \mathcal{C}\rightarrow\mathcal{C}$ given by fixing $I$ in the left argument of $\otimes$ and $Id_\mathcal{C}$ (and a similar story for $\rho$). Is this correct?

Answer 1

The functors under consideration are most apparent if you express the axioms for a monoidal category by commutative diagrams. By analogy, recall that a monoid (in the category of sets) is an object $M$ ogether with an identity map $i:\ast \to M$ from the one-point set $\ast$, together with a multiplication map $\mu: M\times M \to M$ such that certain diagrams commute. One such diagram is the following:

which expresses the fact that $i$ is a left-unital with respect to $\mu$. Here the map $\pi$ is the projection. There is a similar diagram for right-unitality, and one for associativity, namely

Now for a monoidal category $(\mathcal{C}, \otimes, I)$, replace $M$ above by $\mathcal{C}$, $\mu$ by $\otimes$ and $i$ by the constant functor at $I$. Then $\mathcal{C}$ is a monoidal category if the (relabelled) above diagrams commute up to natural isomorphism, and some additional axioms hold (pentagon identity, etc).