8.8.5 Properties of Natural Transformations

Proposition 8.8.5.1.1 ▶ Natural Transformations as Categorical Homotopies

Let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors. The following data are equivalent:[1]

  1. A natural transformation $\alpha \colon F\Longrightarrow G$.
  2. A functor $\webleft [\alpha \webright ]\colon \mathcal{C}\to \mathcal{D}^{\mathbb {1}}$ filling the diagram
  3. A functor $\webleft [\alpha \webright ]\colon \mathcal{C}\times \mathbb {1}\to \mathcal{D}$ filling the diagram

Proof of Proposition 8.8.5.1.1.

From Item 1 to Item 2 and Back
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.

From Item 2 to Item 3 and Back
This follows from Item 3 of Proposition 8.9.1.1.2.

Footnotes

[1] Taken from [MO 64365].
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