Are these identity natural morphisms? So I'm learning about horizontal composition of natural transformations. And I'm looking at this equation.
$$(J \eta)_X: JF \to JG = J\eta_{(X)}$$
It seems to me that what they mean by $J$ in this context is the identity natural transformation $J \to J$. 
But why, instead of writing $J$, don't we write $1_J$, as a identity morphism on the Functor $J$, which seemed to me the common way to notate identity natural transformations up to this point.
Can I exchange $J$ with $1_J$ in my head, or is there an important way, to keep those two apart?
 A: Here's a slightly more general answer to your question.
Given categories and functors as shown:
$$\mathcal{A} \overset{F}{\to} \mathcal{B} \overset{G}{\underset{H}{\rightrightarrows}} \mathcal{C} \overset{K}{\to} \mathcal{D}$$
and a natural transformation $\theta : G \to H$, we can define new natural transformations
$$\theta_F : GF \to HF \quad \text{and} \quad K\theta : KG \to KH$$
by defining
$$(\theta_F)_A = \theta_{F(A)} : GF(A) \to HF(A) \quad \text{and} \quad (K\theta)_B = K(\theta_B) : KG(B) \to KH(B)$$
for all $A \in \mathrm{ob}(\mathcal{A})$ and $B \in \mathrm{ob}(\mathcal{B})$.
These are just definitions of what is meant by '$\theta_F$' and '$K\theta$'; but as you suggest, identity natural transformations are involved: indeed,
$$\theta_F = \theta \star \mathrm{id}_F \quad \text{and} \quad K\theta = \mathrm{id}_K \star \theta$$
where $\star$ is horizontal composition of natural transformations. Horizontal composition is slightly trickier to define, so most authors define $\theta_F$ and $K\theta$ independently.
A: In any category, the only danger of replacing an object $x$ with the identity on it is mild temporary confusion. In fact, there is a formalism of categories in which the are no objects at all, only morphisms, and their composition rule, which is a partial operation on the class of morphisms. You can then consider two categories, the category of all categories as in the definition with objects, and the category of all categories as in the object-free definition. The notion of functor is, of course, suitably adjusted. These two categories are not isomorphic but they are equivalent. This is the categorically precise answer to your question, i.e., yes, you can safely replace an object by its identity. It is convenient to do so as it may reduce clutter in the notation. 
