Variant 8.5.1.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: X \rightarrow X'$ and $g: Y \rightarrow Y'$. The following conditions are equivalent:
- $(1)$
The morphism $g$ is a retract of $f$ in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$.
- $(2)$
The homotopy class $[g]$ is a retract of $[f]$ in the ordinary category $\operatorname{Fun}( [1], \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$.
The implication $(1) \Rightarrow (2)$ is immediate. Conversely, suppose that $(2)$ is satisfied. Then we can choose a commutative diagram
in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, where the horizontal compositions are the identity morphisms $[\operatorname{id}_{Y}]$ and $[ \operatorname{id}_{Y'} ]$, respectively. By virtue of Exercise 1.5.2.10, the squares on the left and right of this diagram can be lifted to commutative diagrams in the $\infty $-category $\operatorname{\mathcal{C}}$, which we can identify with morphisms $\alpha : g \rightarrow f$ and $\beta : f \rightarrow g$ in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. Beware that the composition $(\beta \circ \alpha ): g \rightarrow g$ need not be homotopic to the identity morphism $\operatorname{id}_{g}$. However, the criterion of Theorem 4.4.4.4 guarantees that $\beta \circ \alpha $ is an isomorphism in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. In particular, $\alpha $ admits a left homotopy inverse, and therefore exhibits $g$ as a retract of $f$.