Warning 2.2.1.9. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then there are two different notions of composition for the $2$-morphisms of $\operatorname{\mathcal{C}}$:
- $(V)$
Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$. Suppose we are given $1$-morphisms $f,g,h: X \rightarrow Y$ and a pair of $2$-morphisms
\[ \gamma : f \Rightarrow g \quad \quad \delta : g \Rightarrow h. \]We can then apply the composition law in the ordinary category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ to obtain a $2$-morphism $f \Rightarrow h$, which we refer to as the vertical composition of $\gamma $ and $\delta $.
- $(H)$
Let $X$, $Y$, and $Z$ be objects of $\operatorname{\mathcal{C}}$. Suppose we are given $2$-morphisms $\gamma : f \Rightarrow g$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\gamma ': f' \Rightarrow g'$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)$. Then the image of $(\gamma ', \gamma )$ under the composition law
\[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Z), \]is a $2$-morphism from $f' \circ f$ to $g' \circ g$, which will refer to as the horizontal composition of $\gamma $ and $\gamma '$.
The terminology is motivated by the following graphical representations of the data described in $(V)$ and $(H)$:
To avoid confusion, we will generally denote the vertical composition of $2$-morphisms $\gamma $ and $\delta $ by $\delta \gamma $ and the horizontal composition of $2$-morphisms $\gamma $ and $\gamma '$ by $\gamma ' \circ \gamma $.