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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.5.5.10. Suppose we are given a commutative diagram of $\infty $-categories

7.51
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-as-categorical-limit} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{0} \ar [d] \\ \operatorname{\mathcal{C}}_{1} \ar [r] & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}

which we identify with a functor $\overline{\mathscr {F}}: [1] \times [1] \rightarrow \operatorname{\mathcal{QC}}$. The following conditions are equivalent:

$(1)$

The diagram (7.51) is a categorical pullback square, in the sense of Definition 4.5.2.7.

$(2)$

The functor $\overline{\mathscr {F}}$ is a categorical limit diagram, in the sense of Definition 7.5.5.1.

Proof. Using Proposition 4.5.2.12, we can restate $(2)$ as follows:

$(2')$

For every simplicial set $K$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}_{01})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}(K,\operatorname{\mathcal{C}}_{0})^{\simeq } \ar [d] \\ \operatorname{Fun}(K,\operatorname{\mathcal{C}}_{1})^{\simeq } \ar [r] & \operatorname{Fun}(K,\operatorname{\mathcal{C}})^{\simeq } } \]

is a homotopy pullback square.

The equivalence $(1) \Leftrightarrow (2')$ follows by combining Propositions 7.5.4.13, and 7.5.5.7. $\square$