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Categories for the Working Mathematician says:

Formally, a monoidal category $B = (B, *, e, \alpha,\lambda, \rho)$ is a category $B$, a bifunctor $*: B \times B \to B$, an object $e \in B$, and three natural isomorphisms $\alpha, \lambda, \rho$. Explicitly,

$$ \alpha = \alpha_{a,b,c} : a * (b*c) \equiv (a*b)*c$$

is natural for all $a, b, c \in B$, and $\lambda$ and $\rho$ are natural

$$ \lambda_a: e * a \equiv a$$ $$\rho_a: a*e \equiv a$$

for all objects $a \in B$, and

$$\lambda_e = \rho_e: e * e \to e.$$

$\alpha, \lambda, \rho$ are natural isomorphisms, so they should be natural transformations between functors. But I have difficulty understand they are natural transformations based on how they are used in the definition of a monoidal category above.

  • From what functors to what functors are $\alpha, \lambda, \rho$ respectively?

  • By the definition of a natural transformation, what are the morphisms which $\alpha, \lambda, \rho$ assign to each object respectively?

Thanks.

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  • $\begingroup$ I don't understand your second question; the morphisms $\lambda$ (for instance) assigns to each $a$ is $\lambda_a$. If you're asking how this morphism is defined, that's going to depend on the particular monoidal category; there's no general answer to that. $\endgroup$ – Malice Vidrine Jul 10 at 20:22
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$\alpha$ goes from $-*(-*-)$ to $(-*-)*-$ : these are functors $B^3\to B$, the first one is defined by $(a,b,c)\mapsto a*(b*c)$ and the second one similarly, and on maps well it is defined in the obvious way, knowing that $*: B^2\to B$ is a functor.

$\lambda$ goes from $e*-$ to $id_B$, and similarly for $\rho$

Their specific nature/values depend on the monoidal category in question. If it is a strict monoidal category for instance, they will be the identity. If not, they can be all sorts of things.

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