Kerodon

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Definition 4.5.3.1. A commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

is a categorical pushout diagram if, for every $\infty $-category $\operatorname{\mathcal{C}}$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}(B,\operatorname{\mathcal{C}})^{\simeq } \ar [l] \\ \operatorname{Fun}(C, \operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}(D, \operatorname{\mathcal{C}})^{\simeq } \ar [u] \ar [l] } \]

is a homotopy pullback square.