For natural transformation $\eta$ and functor $L$, is $(\eta \circ L)_X \equiv \eta_{L(X)}$ by definition?

Question

Please see this answer of mine for the source of my confusion.

Highly confused now. I think I had to have gotten that answer right, because the pieces all fit into place and it's very close syntactically.

Answer 1

The simplest way to keep everything straight is to apply Occam's razor to eliminate all of the unnecessary entities.

Everything can be interpreted in the arithmetic of natural transformations, and there are only two operations: vertical composition and horizontal composition.

Objects of a category $\mathcal{C}$ are identified with functors $1 \to \mathcal{C}$. Morphisms are natural transformations between such things.

Arithmetically, functors behave identically to their identity natural transformation.

What you write as $\eta \circ L$ is (presumably) the horizontal composite of $\eta$ with $L$, sometimes called their "whiskering". Explicitly translating everything to natural transformations, it is the horizontal composite $\eta \circ 1_L$.

The "component of a natural transformation" $\eta_X$ is another instance of horizontal composition: it means $\eta \circ X$. Again, if you want a translation, let $\bar{X}$ be the functor corresponding to $X$. Then, $\eta_X$ is the morphism corresponding to $\eta \circ 1_{\bar{X}}$.

The values of a functor? Horizontal composition again: $L(X)$ means $L \circ X$, and $L(f)$ means $L \circ f$.

The equation you are asking about is nothing more than an instance of the fact that horizontal composition is associative:

$$ (\eta \circ L) \circ X = \eta \circ (L \circ X)$$

---

Note, incidentally, that for consistency, when you use $\eta \circ L$ for horizontal composition, you should also use $F \circ G$ for composition of functors, and you should use a different symbol entirely for composing morphisms and for vertical composition of natural transformations; e.g. $f \cdot g$ and $\eta \cdot \mu$.

I emphasize this because some sources use alternative notation convention where juxtaposition is used for horizontal composition and $\circ$ is reserved for vertical composition. In this notation convention, the equation you wrote would be written $(\eta L)_X = \eta_{L(X)}$.