Remark 4.5.2.9. Suppose we are given a categorical pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}. } \]
Then, for every simplicial set $X$, the induced diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(X, \operatorname{\mathcal{C}}_{01} ) \ar [r] \ar [d] & \operatorname{Fun}(X, \operatorname{\mathcal{C}}_0) \ar [d] \\ \operatorname{Fun}(X, \operatorname{\mathcal{C}}_1) \ar [r] & \operatorname{Fun}( X, \operatorname{\mathcal{C}}) } \]
is also a categorical pullback square. This follows by combining Remarks 4.5.2.6 and 4.5.1.16.