Warning 7.6.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a morphism, which we depict as a diagram
Beware that, if $\sigma $ is a pullback square in the $\infty $-category $\operatorname{\mathcal{C}}$, then the associated diagram
need not be a pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (see Example 7.6.3.4 and Exercise 7.6.3.5). If $Y$ is an object of $\operatorname{\mathcal{C}}$, then the map of sets
is surjective, but need not be injective. Given a commutative diagram
in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, we can always find a morphism $g_{01}: Y \rightarrow X_{01}$ satisfying $g_{01}: Y \rightarrow X_{01}$ satisfying $[g_0] = [f'_0] \circ [g_{01}]$ and $[ g_1 ] = [ f'_1 ] \circ [ g_{01} ]$. However, the homotopy class $[g_{01}]$ is not uniquely determined: roughly speaking, to construct $g_{01}$, we need to lift (7.57) to a commutative diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Such a lift always exists (Exercise 1.5.2.10), but is not unique (even up to homotopy).