Remark 2.2.1.10. Let $\operatorname{\mathcal{C}}$ be a $2$-category. For each object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the identity $1$-morphism $\operatorname{id}_{X}$ and the unit constraint $\upsilon _{X}$ are determined (up to unique isomorphism) by the composition law and associativity constraints. More precisely, given any other choice of identity morphism $\operatorname{id}'_{X}$ and unit constraint $\upsilon '_{X}: \operatorname{id}'_{X} \circ \operatorname{id}'_{X} \xRightarrow {\sim } \operatorname{id}'_{X}$, there exists a unique invertible $2$-morphism $\gamma : \operatorname{id}_{X} \xRightarrow {\sim } \operatorname{id}'_{X}$ for which the diagram
commutes. This follows from Proposition 2.1.2.9, applied to the monoidal category $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ of Remark 2.2.1.7.
It is possible to adopt a variant of Definition 2.2.1.1 where we do not require the identity morphisms $\{ \operatorname{id}_ X \} _{X \in \operatorname{Ob}(\operatorname{\mathcal{C}})}$ (or unit constraints $\{ \upsilon _{X} \} _{X \in \operatorname{Ob}(\operatorname{\mathcal{C}})}$) to be explicitly specified. This variant is equivalent to Definition 2.2.1.1 for many purposes. However, it is not suitable for our applications: in ยง2.3, we associate to each $2$-category $\operatorname{\mathcal{C}}$ a simplicial set $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ called the Duskin nerve of $\operatorname{\mathcal{C}}$, whose degeneracy operators depend on the choice of identity morphisms and unit constraints in $\operatorname{\mathcal{C}}$ (though the face operators do not: see Warning 2.3.1.11).