Remark 7.7.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits an initial object $\emptyset $. Then, for every pair of morphisms $f_{-}: X_{-} \rightarrow X$ and $f_{+}: X_{+} \rightarrow X$, we can choose a commutative diagram $\sigma :$
in $\operatorname{\mathcal{C}}$. Moreover, the diagram $\sigma $ is essentially unique: it is characterized up to isomorphism by the requirement that it is a left Kan extension of the subdiagram obtained by omitting the upper left hand corner. Consequently, we can replace condition $(2)$ of Definition 7.7.4.1 by either of the following:
- $(2')$
For every pair of distinct elements $i,j \in I$, there exists a pullback diagram
\[ \xymatrix { \emptyset \ar [r] \ar [d] & X_ i \ar [d]^{f_ i} \\ X_ j \ar [r]^{f_ j} & X } \]in the $\infty $-category $\operatorname{\mathcal{C}}$.
- $(2'')$
For every pair of distinct elements $i,j \in I$, every commutative diagram
\[ \xymatrix { \emptyset \ar [r] \ar [d] & X_ i \ar [d]^{f_ i} \\ X_ j \ar [r]^{f_ j} & X } \]is a pullback diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.