Remark 7.7.2.21. Let $K$ be a simplicial set, let $\operatorname{\mathcal{C}}$ be an let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and $K$-indexed colimits, and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Stated more informally, Remark 7.7.2.20 asserts that that the colimit of $F$ is strongly universal if and only if the following conditions are satisfied:
- $(a')$
For every morphism $C' \rightarrow C = \varinjlim (F)$ in $\operatorname{\mathcal{C}}$, the comparison map
\[ \varinjlim _{x \in K}( C' \times _{C} F(x) ) \rightarrow C' \times _{C} \varinjlim _{x \in K}( F(x) ) \simeq C' \]is an isomorphism.
- $(b')$
For every cartesian natural transformation $\gamma : F' \rightarrow F$, the diagram
\[ \xymatrix { F'(x) \ar [r] \ar [d] & \varinjlim (F) \ar [d] \\ F(x) \ar [r] & \varinjlim (F) } \]is a pullback square for each vertex $x \in K$.