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Remark Let $\operatorname{\mathcal{C}}$ be a strictly unitary $2$-category. Then $\operatorname{\mathcal{C}}$ satisfies the following stronger versions of conditions $(a)$ and $(c)$ of Proposition


For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functors

\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \quad \quad f \mapsto \operatorname{id}_{Y} \circ f \]
\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \quad \quad f \mapsto f \circ \operatorname{id}_ X \]

are equal to the identity.


For every pair of $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in $\operatorname{\mathcal{C}}$, the associativity constraints $\alpha _{g, f, \operatorname{id}_ X}$, $\alpha _{ g, \operatorname{id}_ Y, f}$, and $\alpha _{\operatorname{id}_ Z, g, f}$ are equal to the identity (as $2$-morphisms from $g \circ f$ to itself).

Here $(a')$ follows from the naturality of the left and right unit constraints (Remark, and $(c')$ follows from Propositions and