Example 9.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be an isomorphism in $\operatorname{\mathcal{C}}$. Let $F$ denote the composite map $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )^{\triangleright } \twoheadrightarrow (\Delta ^{0})^{\triangleright } \simeq \Delta ^1 \xrightarrow {f} \operatorname{\mathcal{C}}$, which we display informally as a diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [drr]^-{f} \ar [r]^-{\operatorname{id}} & X \ar [dr]^-{f} \ar [r]^-{\operatorname{id}} & X \ar [d]^-{f} \ar [r]^-{\operatorname{id}} & X \ar [dl]^-{f} \ar [r]^-{\operatorname{id}} & \cdots \\ & & Y. & & } \]
Since the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ is contractible (Example 3.2.4.2), the functor $F$ is a colimit diagram (Corollary 7.2.3.5), and therefore exhibits $f$ as a transfinite composition of morphisms belonging to the singleton $\{ \operatorname{id}_{X} \} $.