Definition 1.6 (Natural transformation). Given categories $C$ and $D$, and two functors $C\underset{G}{\overset{F}{\rightrightarrows }}D$, a natural transformation $\alpha :F\to G$ assigns to each $A\in \mathrm{ob}C$ a morphism ${\alpha }_A:FA\to GA$ in $D$, such that for any $A\stackrel{f}{\to }B$ in $C$, the square

commutes (we call this square the naturality square for $\alpha$ at $f$). Given $\alpha$ as above, and $\beta :G\to H$, we define $\beta \alpha :F\to H$ by ${(\beta \alpha )}_A={\beta }_A{\alpha }_A$. We write $[C,D]$ for the category of functors $C\to D$ and natural transformations between them.