The vertical composition of two natural transformations $\alpha \colon F\Longrightarrow G$ and $\beta \colon G\Longrightarrow H$ as in the diagram
is the natural transformation $\beta \circ \alpha \colon F\Longrightarrow H$ consisting of the collection
\[ \webleft\{ \webleft (\beta \circ \alpha \webright )_{A} \colon F\webleft (A\webright ) \to H\webleft (A\webright ) \webright\} _{A\in \text{Obj}\webleft (\mathcal{C}\webright )} \]
with
\[ \webleft (\beta \circ \alpha \webright )_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\beta _{A}\circ \alpha _{A} \]
for each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$.
