Example 4.3.6.15 (Composition Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, which we identify with a diagram $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. The inclusions $\{ 0\} \hookrightarrow \Delta ^1 \hookleftarrow \{ 1\} $ then induce restriction functors $\operatorname{\mathcal{C}}_{X/} \xleftarrow { e_0 } \operatorname{\mathcal{C}}_{f/} \xrightarrow { e_1 } \operatorname{\mathcal{C}}_{Y/}$. It follows from Corollary 4.3.6.14 that $e_1$ is a trivial Kan fibration, and therefore admits a section $s: \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}_{f/}$ (which is unique up to isomorphism). The composition $e_0 \circ s$ can then be viewed as a functor from $\operatorname{\mathcal{C}}_{Y/}$ to $\operatorname{\mathcal{C}}_{X/}$, which we will refer to as precomposition with $f$. Concretely, this functor takes an object $g: Y \rightarrow Z$ of the $\infty $-category $\operatorname{\mathcal{C}}_{Y/}$ to an object $h: X \rightarrow Z$ of the $\infty $-category $\operatorname{\mathcal{C}}_{X/}$, which is characterized (up to isomorphism) by the requirement that there exists a $2$-simplex
so that $h$ is a composition of $f$ with $g$ in the sense of Definition 1.4.4.1. Applying the same construction in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$, we obtain a functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ which we will refer to as postcomposition with $f$; concretely, it carries an object $e: W \rightarrow X$ of the $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ to an object $W \rightarrow Y$ of $\operatorname{\mathcal{C}}_{/Y}$ which is a composition of $e$ with $f$.