# In the definition of a monoidal category, how are the natural isomorphisms natural transformations?

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.

• 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. – Malice Vidrine Jul 10 at 20:22

## 1 Answer

$$\alpha$$ 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.