# Kerodon

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Example 7.5.2.15. Let $\operatorname{\mathcal{K}}$ be the partially ordered set depicted in the diagram

$\bullet \rightarrow \bullet \leftarrow \bullet$

and suppose we are given a functor $\mathscr {F}: \operatorname{\mathcal{K}}\rightarrow \operatorname{QCat}$, which we depict as a diagram of $\infty$-categories

$\operatorname{\mathcal{C}}_0 \xrightarrow {T_0} \operatorname{\mathcal{C}}\xleftarrow {T_1} \operatorname{\mathcal{C}}_1.$

Then the homotopy limit $\underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ can be identified with the iterated homotopy pullback $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} (\operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}})$. Applying Corollary 4.5.2.18, we see that the equivalence $\operatorname{\mathcal{C}}_1 \hookrightarrow \operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}$ of Example 7.5.2.14 induces an equivalence of $\infty$-categories

$\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \hookrightarrow \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} (\operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}) \simeq \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}).$

In particular, the comparison map $\varprojlim (\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is a categorical equivalence of simplicial sets if and only if the inclusion $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \hookrightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is a categorical equivalence of simplicial sets. This condition is satisfied if either $T_0$ or $T_1$ is a isofibration of $\infty$-categories (Corollary 4.5.2.22), but not in general.