Remark 2.2.7.6. Let $\operatorname{\mathcal{C}}$ be a strictly unitary $2$-category, let $\{ \mu _{g,f} \} $ be a twisting cochain for $\operatorname{\mathcal{C}}$ (see Notation 2.2.6.7), and let $\operatorname{\mathcal{C}}'$ denote the twist of $\operatorname{\mathcal{C}}'$ with respect to $\{ \mu _{g,f} \} $ (Construction 2.2.6.8). The following conditions are equivalent:
- $(1)$
The $2$-category $\operatorname{\mathcal{C}}'$ is strictly unitary.
- $(2)$
For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, both $\mu _{f, \operatorname{id}_ X}$ and $\mu _{\operatorname{id}_{Y},f}$ are identity $2$-morphisms (from $f \circ \operatorname{id}_{X} = f = \operatorname{id}_{Y} \circ f$ to itself).
If these conditions are satisfied, we will say that the twisting cochain $\{ \mu _{g,f} \} $ is normalized.