Subsection 9.9.1 (017L): Transformations

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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

\[ \webleft\{ \alpha _{A}\colon F\webleft (A\webright )\to G\webleft (A\webright )\webright\} _{A\in \text{Obj}\webleft (\mathcal{C}\webright )} \]

of morphisms of $\mathcal{D}$.

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

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

We have an isomorphism

\[ \text{Trans}\webleft (F,G\webright )\cong \prod _{A\in \mathcal{C}}\textup{Hom}_{\mathcal{D}}\webleft (F_{A},G_{A}\webright ). \]

Proof of Remark 9.9.1.1.3.

Clear.

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