Difficulty understanding natural transformation notation

Question

I'm trying to fully understand the conditions for a monad as they are written in category theory. Left and right identity are expressed as follows: $$\mu \circ \eta T = \mu \circ T \eta = id_T$$ Where $C$ is a category, $T:C \rightarrow C$, $\mu:T\circ T \rightarrow T$ and $\eta : id_C \rightarrow T$. If I understand correctly, $$\eta T : id_C \circ T \rightarrow T \circ T$$ $$T \eta : T \circ id_C \rightarrow T \circ T$$ But $T$ left or right composed with $id_C$ should just be $T$... which would mean that $\eta T$ and $T\eta$ have the same type signature. What distinguishes them?

Answer 1

I think it helps a lot to translate other concepts into this algebraic notation. For example:

• Use juxtaposition $Fx$ and $Ff$ to give the value of a functor $F$ at an object $x$ or a morphism $f$
• Use juxtaposition $\eta x$ to give the component of a natural transformation $\eta$ at the object $x$

In fact, if you express an object as a functor $1 \to \mathcal{C}$ and an arrow as a natural transformation between such functors, then the above isn't even a translation; the horizontal composite $\eta x$ is literally the arrow corresponding to $\eta_x$.

Since we usually understand natural transformations by their action on objects, we can see their difference: the products involved in the two formulas $\eta T x$ and $T \eta x$ are in different orders. Using this to help compute the components of both natural transformations in more familiar notation:

• $(\eta T)_x = \eta T x = \eta T(x) = \eta_{T(x)} $
• $(T \eta)_x = T \eta x = T \eta_x = T(\eta_x) $

Answer 2

Yes, $\eta T, T\eta : T\to T\circ T$. However, $(\eta T)_x = \eta_{Tx}$, whereas $(T\eta)_x = T(\eta_x)$ so a priori there's no reason they should be equal for an arbitrary functor.

In particular, this is true for a monad.

If you want to understand (or at least easily remember) the axioms for a monad, try to think of them as saying "$T$ is a monad if and only if it is a monoid in the monoidal category $(C^C, \circ)$"