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.
$\alpha$ 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.