Definition 1.2.8 (Natural Transformations).label Let $F,G:\frak{C}\to\frak{D}$ be covariant functors. A natural transformation or morphism of functors $\al$ from $F$ to $G$ is a rule which associates each object $X\in \ob{\frak{C}}$ with a morphism $\al_{X}:F(X)\to G(X)$ such that for each morphism $f\in \mor{\frak{C}}{X }{Y }$ we have a commutative diagram:
\[\xymatrix{ F(X) \ar@{->}[rr]^{\alpha_X} \ar@{->}[dd]^{F(f)} & & G(X) \ar@{->}[dd]^{G(f)} \\ & & \\ F(Y) \ar@{->}[rr]^{\alpha_Y} & & G(Y) }\]
If each $\al_{X}$ is an isomorphism then $F$ and $G$ are naturally equivalent or isomorphic, denoted by $F\iso G$. In which case
\begin{align*}\mor{\frak{D}}{F(X)}{F(Y)}\iso \mor{\frak{D}}{G(X)}{G(Y)}&&g\mto \al_{B}\circ g\circ\al_{X}\inv\end{align*}
A similar definition can be made for contravariant functors by observation in Definition 1.2.7.
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