Let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors. The following data are equivalent:1
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1.
A natural transformation $\alpha \colon F\Longrightarrow G$.
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2.
A functor $\webleft [\alpha \webright ]\colon \mathcal{C}\to \mathcal{D}^{\mathbb {1}}$ filling the diagram
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3.
A functor $\webleft [\alpha \webright ]\colon \mathcal{C}\times \mathbb {1}\to \mathcal{D}$ filling the diagram
11.9.6 Properties of Natural Transformations
Proof of Proposition 11.9.6.1.1.
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
This follows from Item 3 of Proposition 11.10.1.1.2.
