Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 4.4.1.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms in $\operatorname{\mathcal{C}}$. Then the subcategory $\operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ is replete. Unwinding the definitions, this amounts to the observation that for every commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{u} \ar [d]^{v} & Y \ar [d]^{v'} \\ X' \ar [r]^-{u'} & Y' } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$ where $u$, $v$, and $v'$ are isomorphisms, the morphism $u'$ is also an isomorphism. This follows immediately from the two-out-of-three property of Remark 1.3.6.3.