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Proposition 7.1.6.3. 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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{f/} \ar [r] \ar [d]^{F_{f/}} & \{ f\} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K^{\triangleright } ) \ar [d]^{U''} \\ \operatorname{\mathcal{D}}_{ (F \circ f)/} \ar [r] & \{ F \circ f \} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}), } \]

where the horizontal maps are equivalences of $\infty $-categories (see Remark 4.6.4.20). Applying Remark 7.1.4.9, 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 5.3.7.3, 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 7.1.4.19, that $\overline{f}$ is $U''$-initial (when regarded as an object of $\{ f\} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K^{\triangleright } )$) if and only if it is $U'$-initial (when viewed as an object of $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$). $\square$