Proposition 7.1.7.4. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $K$ be a simplicial set, and let
\[ U': \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) \]
be the restriction map. Then a morphism of simplicial sets $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram if and only if it is $U'$-initial when viewed as an object of the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$.
Proof.
Set $f = \overline{f}|_{K}$, so that $U'$ restricts to a functor
\[ U'': \{ f \} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \{ U \circ f \} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}). \]
We have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{f/} \ar [r] \ar [d]^{F_{f/}} & \{ f\} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \ar [d]^{U''} \\ \operatorname{\mathcal{D}}_{ (F \circ f)/} \ar [r] & \{ F \circ f \} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}), } \]
where the horizontal maps are equivalences of $\infty $-categories (see Example 4.6.6.8). Applying Remark 7.1.5.9, we see that $\overline{f}$ is an $U$-colimit diagram if and only if it is $U''$-initial when viewed as an object of the fiber $\{ f\} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K^{\triangleright } )$.
We have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \ar [rr]^{U'} \ar [dr]^{V} & & \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) \ar [dl]^{V'} \\ & \operatorname{Fun}(K, \operatorname{\mathcal{C}}). & } \]
Applying Corollary 5.3.7.4, we see that $V$ and $V'$ are cartesian fibrations and that $U'$ carries $V$-cartesian morphisms of $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ to $V'$-cartesian morphisms of $\operatorname{Fun}(K,\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}})$. It follows from Proposition 7.1.5.19, that $\overline{f}$ is $U''$-initial (when regarded as an object of $\{ f\} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$) if and only if it is $U'$-initial (when viewed as an object of $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$).
$\square$