$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.3.9.5. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories, and suppose we are given a lifting problem
7.44
\begin{equation} \begin{gathered}\label{equation:relative-colimit-existence-criterion} \xymatrix@R =50pt@C=50pt{ K \ar [r]^-{f_0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ K^{\triangleright } \ar [r]^-{g } \ar@ {-->}[ur]^{ \overline{f}_0 } & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}
Let $v \in K^{\triangleright }$ be the cone point and set $D = g(v)$. Then there exists a diagram $f_1: K \rightarrow \operatorname{\mathcal{C}}_{D} \subseteq \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : f_0 \rightarrow f_1$ which carries each vertex $x \in K$ to a $U$-cocartesian morphism $\alpha _ x: f_0(x) \rightarrow f_1(x)$ of $\operatorname{\mathcal{C}}$, where$U \circ \alpha $ is given by the composition $\Delta ^1 \times K \xrightarrow {c} K^{\triangleright } \xrightarrow {g} \operatorname{\mathcal{D}}$. Moreover, the lifting problem (7.44) admits a solution $\overline{f}_0: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which is a $U$-colimit diagram if and only if the following pair of conditions is satisfied:
- $(1)$
The diagram $f_1$ admits a colimit $\overline{f}_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$.
- $(2)$
Let $e: D \rightarrow D'$ be a morphism in the $\infty $-category $\operatorname{\mathcal{D}}$ and let $e_{!}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}_{D'}$ be the covariant transport functor of Notation 5.2.2.9. Then $(e_{!} \circ \overline{f}_1): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D'}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D'}$.
Proof.
The existence (and essential uniqueness) of the diagram $f_1$ and the natural transformation $\alpha : f_0 \rightarrow f_1$ follow from Proposition 5.2.1.3. Let us first show that conditions $(1)$ and $(2)$ are necessary. Suppose that the lifting problem (7.44) admits a solution $\overline{f}_0: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which is a $U$-colimit diagram. Using Proposition 5.2.1.3, we can extend $f_1$ to a diagram $\overline{f}_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ and $\alpha $ to a natural transformation $\overline{\alpha }: \overline{f}_0 \rightarrow \overline{f}_1$ which carries each vertex $x \in K^{\triangleright }$ to a $U$-cocartesian morphism $\overline{\alpha }_ x: \overline{f}_0(x) \rightarrow \overline{f}_1(x)$. Proposition 7.3.9.1 guarantees that $\overline{f}_{1}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, and therefore satisfies conditions $(1)$ and $(2)$ by virtue of Proposition 7.3.9.2.
Now suppose that conditions $(1)$ and $(2)$ are satisfied. Let $\overline{f}_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ be a colimit diagram extending $f_1$. It follows from $(2)$ that $\overline{f}_1$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Let $\pi : ( \Delta ^1 \times K )^{\triangleright } \rightarrow K^{\triangleright }$ denote the morphism which is the identity when restricted to $\{ 0\} \times K$, and which carries $(\{ 1\} \times K)^{\triangleright }$ to the cone point of $K^{\triangleright }$. Since the inclusion map $\{ 1 \} \times K \hookrightarrow \Delta ^1 \times K$ is right cofinal (Proposition 7.2.1.3), Proposition 7.2.2.9 guarantees that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times K \ar [r]^-{\alpha } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ (\Delta ^1 \times K)^{\triangleright } \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur]^{\overline{\alpha }} & \operatorname{\mathcal{D}}} \]
admits a solution $\overline{\alpha }: (\Delta ^1 \times K)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which is a $U$-colimit diagram. Note that in this case $\overline{f}'_1 = \overline{\alpha }|_{ (\{ 1\} \times K)^{\triangleright } }$ is also a $U$-colimit diagram (Corollary 7.2.2.2). Setting $\overline{f}_0 = \overline{\alpha }|_{ (\{ 0\} \times K)^{\triangleright } }$, we note that $\overline{\alpha }$ determines a natural transformation of functors $\overline{f}_0 \rightarrow \overline{f}'_1$ which carries each vertex of $x$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$ and carries the cone point to an identity morphism of $\operatorname{\mathcal{C}}$. Applying the criterion of Proposition 7.3.9.1, we conclude that $\overline{f}_0$ is a $U$-colimit diagram which solves the lifting problem (7.44).
$\square$