The horizontal composition[1][2] of two natural transformations $\alpha \colon F\Longrightarrow G$ and $\beta \colon H\Longrightarrow K$ as in the diagram
of $\alpha $ and $\beta $ is the natural transformation
\[ \beta \mathbin {\star }\alpha \colon \webleft (H\circ F\webright )\Longrightarrow \webleft (K\circ G\webright ), \]
as in the diagram
consisting of the collection
\[ \webleft\{ \webleft (\beta \mathbin {\star }\alpha \webright )_{A} \colon H\webleft (F\webleft (A\webright )\webright ) \to K\webleft (G\webleft (A\webright )\webright ) \webright\} _{A\in \text{Obj}\webleft (\mathcal{C}\webright )}, \]
of morphisms of $\mathcal{E}$ with
