Proposition 2.2.1.16. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing a pair of composable $1$-morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$. Then:
- $(1)$
The associativity constraint $\alpha _{\operatorname{id}_ Z,g,f}: \operatorname{id}_{Z} \circ (g \circ f) \Rightarrow (\operatorname{id}_ Z \circ g) \circ f$ is given by the (vertical) composition
\[ \operatorname{id}_{Z} \circ (g \circ f) \xRightarrow { \lambda _{g \circ f} } g \circ f \xRightarrow {\lambda ^{-1}_{g} \circ \operatorname{id}_{f} } (\operatorname{id}_ Z \circ g) \circ f. \]- $(2)$
The associativity constraint $\alpha _{g,f,\operatorname{id}_ X}: g \circ (f \circ \operatorname{id}_ X) \Rightarrow (g \circ f) \circ \operatorname{id}_ X$ is given by the (vertical) composition
\[ g \circ (f \circ \operatorname{id}_ X) \xRightarrow { \operatorname{id}_{g} \circ \rho _{f}} g \circ f \xRightarrow { \rho ^{-1}_{g \circ f} } (g \circ f) \circ \operatorname{id}_ X \]