# Kerodon

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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 $(H)$:

$\xymatrix@C =50pt{ X \ar@ /^30pt/[r]^{f} \ar [r]^-{g} \ar@ /_30pt/[r]_{h} \ar@ {=>}[]+<32pt,20pt>;+<32pt,10pt>^{\gamma } \ar@ {=>}[]+<32pt,-10pt>;+<32pt,-20pt>^{\delta } & Y & X \ar@ /^15pt/[r]^{f} \ar@ /_15pt/[r]_{g} \ar@ {=>}[]+<32pt,10pt>;+<32pt,-10pt>^{\gamma } & Y \ar@ /^15pt/[r]^{f'} \ar@ /_15pt/[r]_{g'} \ar@ {=>}[]+<32pt,10pt>;+<32pt,-10pt>^{\gamma '}& Z. }$

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 $\gamma ' \circ \gamma$.