Let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors. The following data are equivalent:[1]
Proof of Proposition 8.8.5.1.1.
We may identify $\mathcal{D}^{\mathbb {1}}$ with $\mathsf{Arr}(\mathcal{D})$. Given a natural transformation $\alpha \colon F\Longrightarrow G$, we have a functor making the diagram in Item 2 commute. Conversely, every such functor gives rise to a natural transformation from $F$ to $G$, and these constructions are inverse to each other.
This follows from Item 3 of Proposition 8.9.1.1.2.