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Proposition 4.6.9.11 (Unitality). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X$ and $Y$. Then:
- $(1)$
The composition
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \{ \operatorname{id}_{X} \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]
is equal to the identity (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).
- $(2)$
The composition
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \{ \operatorname{id}_{Y} \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]
is equal to the identity (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).
Proof.
There is a diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X,Y) \ar [dr] \ar [d] & \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{ \operatorname{id}\times \{ \operatorname{id}_{X} \} } \ar [ur] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \ar@ {-->}[r]^-{\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), } \]
where the left diagonal arrow is induced by the map $\sigma ^0_{1}: [2] \rightarrow [1]$ of Construction 1.1.2.1 and the right diagonal arrow is induced by the map $\delta ^{1}_{2}: [1] \rightarrow [2]$ of Construction 1.1.1.4. Here the solid arrows are well-defined as morphisms of simplicial sets, while the dotted arrow is well-defined only as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We now observe that the triangle on the left is strictly commutative, the triangle on the right commutes up to homotopy (by the construction of the composition law $\circ $). Assertion $(1)$ follows from the observation that the composition of the diagonal arrows is the identity on the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (since $\sigma ^{0}_{1} \circ \delta ^{1}_{2}$ is the identity on the object $[1] \in \operatorname{{\bf \Delta }}$). Assertion $(2)$ follows by a similar argument.
$\square$