\chapter{Natural transformations}

Category theory has the potential to get infinitely metamathematical; Functors are morphisms of the $\mathbf{Cat}$ category. Imagine the category of functors between $\mathcal{C}$ to $\mathcal{D}$, then \emph{natural transformations} are the functors of the functors!


functors between the same categories

\begin{definition}[Natural transformation]
	\emph{natural transformation} between 2 functors $F,G : \mathcal{C} \to \mathcal{D}$ is family of morphisms (component mappings) $(\alpha_{X})_{X \in \mathrm{Obj}(\mathcal{D})}$ of the form $\alpha_X : F(X) \to G(X)$ that satisfy the following commutative diagram for any $f \in \mathrm{Hom}(X,Y)$.

% https://q.uiver.app/#q=WzAsNCxbMSwxLCJHKFkpIl0sWzAsMCwiRihYKSJdLFsxLDAsIkYoWSkiXSxbMCwxLCJHKFgpIl0sWzEsMiwiRihmKSJdLFszLDAsIkcoZikiLDJdLFsxLDMsIlxcYWxwaGFfWCIsMl0sWzIsMCwiXFxhbHBoYV9ZIl1d
\[\begin{tikzcd} {F(X)} & {F(Y)} \\ {G(X)} & {G(Y)} \arrow["{F(f)}", from=1-1, to=1-2] \arrow["{\alpha_X}"', from=1-1, to=2-1] \arrow["{\alpha_Y}", from=1-2, to=2-2] \arrow["{G(f)}"', from=2-1, to=2-2] \end{tikzcd}\]

\end{definition}


For any 2 functors $F: \mathcal{C} \to  \mathcal{D}$ and $G : \mathcal{C} \to \mathcal{D}$, a natural transformation is a family of morphisms  of D mu such that for any morphism $f :X \to Y$ of $\mathcal{C}$, we have mu satisfying the following

$\mu_X \circ F(f) = G(f) \circ \mu_Y$
\section{Natural transformation}
\section{Yoneda's lemma}