Remark 11.4.0.20. This tag was removed because the definition of $2$-category was changed.
Let $\operatorname{\mathcal{C}}$ be a strictly unitary $2$-category. Then the triangle identity $(T)$ of Definition 2.2.1.1 can be stated more simply as the assertion that for every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, the associativity constraint $\alpha _{g, \operatorname{id}_ Y, f}$ coincides with the identity map from $g \circ (\operatorname{id}_{Y} \circ f) = g \circ f = (g \circ \operatorname{id}_{Y} ) \circ f$ to itself.