Kerodon

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

Remark 1.4.1.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. If $f,g: X \rightarrow Y$ are homotopic morphisms of $\operatorname{\mathcal{C}}$, then $F(f), F(g): F(X) \rightarrow F(Y)$ are homotopic morphisms of $\operatorname{\mathcal{D}}$. More precisely, the functor $F$ carries homotopies from $f$ to $g$ (viewed as $2$-simplices of $\operatorname{\mathcal{C}}$) to homotopies from $F(f)$ to $F(g)$ (viewed as $2$-simplices of $\operatorname{\mathcal{D}}$).