Example 10.3.1.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r]^-{u} & Y, } \]
and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be the sieve on $Y$ generated by the morphism $f$. Then the pullback $u^{\ast }( \operatorname{\mathcal{C}}^{0}_{/Y} )$ is the sieve on $Y'$ generated by the morphism $f'$. In other words, a morphism $[v]: C \rightarrow X'$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ factors through $[f']$ if and only if the composite morphism $[u] \circ [v]$ factors through $[f]$ (see Warning 7.6.2.3).