Notation 11.9.1.1.2 (017N): The Set of Transformations Between Two Functors

Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors.

A transformation¹ $\smash {\alpha \colon F\Rightarrow G}$ from $F$ to $G$ is a collection

\[ \left\{ \alpha _{A}\colon F(A)\to G(A)\right\} _{A\in \operatorname {\mathrm{Obj}}(\mathcal{C})} \]

of morphisms of $\mathcal{D}$.

¹Further Terminology: Also called an unnatural transformation for emphasis.

We write $\operatorname {\mathrm{Trans}}(F,G)$ for the set of transformations from $F$ to $G$.

We have an equality

\[ \operatorname {\mathrm{Trans}}(F,G)=\prod _{A\in \mathcal{C}}\operatorname {\mathrm{Hom}}_{\mathcal{D}}(F_{A},G_{A}). \]

Proof of Remark 11.9.1.1.3.

This follows by definition.

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