Remark 2.2.7.3. 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 2.2.7.2:
- $(a')$
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.
- $(c')$
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 2.2.1.13), and $(c')$ follows from Propositions 2.2.1.14 and 2.2.1.16.