Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

7.1.8 Limits and Colimits of Functors

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $B$ be a simplicial set. For every vertex $b \in B$, we let

\[ \operatorname{ev}_{b}: \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ b\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}} \]

denote the functor given by evaluation at $b$. Our goal in this section is to show that the collection of functors $\{ \operatorname{ev}_ b \} _{b \in B}$ creates colimits in the following sense:

Proposition 7.1.8.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $f: K \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the composite diagram

\[ K \xrightarrow {f} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]

admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}$. Then:

$(1)$

The diagram $f$ admits a colimit in $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$.

$(2)$

Let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be an extension of $f$. Then $\overline{f}$ is a colimit diagram if and only if, for every vertex $b \in B$, the morphism

\[ K^{\triangleright } \xrightarrow { \overline{f} } \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]

is a colimit diagram in $\operatorname{\mathcal{C}}$.

Corollary 7.1.8.2. Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $K$-indexed colimits. Then, for every simplicial set $B$, the $\infty $-category $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$ also admits $K$-indexed colimits. Moreover, a morphism of simplicial sets $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ is a colimit diagram if and only if, for every vertex $b \in B$, the morphism

\[ K^{\triangleright } \xrightarrow { \overline{f} } \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]

is a colimit diagram in $\operatorname{\mathcal{C}}$.

We will give a proof of Proposition 7.1.8.1 at the end of this section. Our strategy is to deduce Proposition 7.1.8.1 from a pair of more general results which apply to relative colimit diagrams (Corollaries 7.1.8.7 and 7.1.8.11). The increased flexibility of the relative setting will allow us to reduce to the case $K = \emptyset $, by virtue of the following:

Proposition 7.1.8.3. 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 7.1.6.8, 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 5.1.7.3, 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 7.1.6.15, 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 $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$). $\square$

Remark 7.1.8.4. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets having restriction $f = \overline{f}|_{K}$. Proposition 7.1.8.3 asserts that $\overline{f}$ is a $U$-colimit diagram if and only if, for every diagram $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ having restriction $g = \overline{g}|_{K}$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) }( \overline{f}, \overline{g} ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f, g ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{D}}) }( U \circ \overline{f}, U \circ \overline{g} ) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f, U \circ g) } \]

is a homotopy pullback square. However, it suffices to verify this condition in the special case where $\overline{g}$ is a constant diagram: that is the content of Proposition 7.1.7.12.

Corollary 7.1.8.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let

\[ U: \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \]

denote the restriction map. Then a morphism of simplicial sets $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a 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. Apply Proposition 7.1.8.3 in the special case $\operatorname{\mathcal{D}}= \Delta ^{0}$. $\square$

Corollary 7.1.8.6. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $B$ and $K$ be simplicial sets, and let $A \subseteq B$ be a simplicial subset which contains every vertex of $B$. Suppose we are given a lifting problem

7.7
\begin{equation} \begin{gathered}\label{equation:relative-colimit-pointwise-existence-strong} \xymatrix@R =50pt@C=50pt{ (B \times K) \coprod _{ (A \times K ) } (A \times K^{\triangleright }) \ar [r]^-{ f } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ B \times K^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{f} } & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

which satisfies the following condition:

$(\ast )$

Let $\sigma : \Delta ^ n \rightarrow B$ be an $n$-simplex which does not belong to $A$, and let $a = \sigma (0)$ be the initial vertex. Then the restriction

\[ f_{a} = f|_{ \{ a\} \times K^{\triangleright } }: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}} \]

is a $U$-colimit diagram.

Then the lifting problem (7.7) admits a solution $\overline{f}: B \times K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

Proof. Set $\operatorname{\mathcal{C}}' = \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ and $\operatorname{\mathcal{D}}' = \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}})$, so that $U$ induces an inner fibration $U': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}'$ (Proposition 4.1.4.1). We can then rewrite (7.7) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{g} \ar [d] & \operatorname{\mathcal{C}}' \ar [d]^{U'} \\ B \ar@ {-->}[ur] \ar [r]^-{g_0} & \operatorname{\mathcal{D}}'. } \]

Let $P$ be the partially ordered set of pairs $(A', g')$, where $A' \subseteq B$ is a simplicial subset containing $A$, and $g': A' \rightarrow \operatorname{\mathcal{C}}'$ is a morphism satisfying $g'|_{A} = g$ and $U' \circ g' = g_0|_{A'}$. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma and therefore contains a maximal element $(A_{\mathrm{max}}, g_{\mathrm{max}})$. To complete the proof, it will suffice to show that $A_{\mathrm{max}} = B$. Assume otherwise: then there exists some $n$-simplex $\sigma : \Delta ^ n \rightarrow B$ which is not contained in $A_{\mathrm{max}}$. Choose $n$ as small as possible, so that $\sigma $ carries the boundary $\operatorname{\partial \Delta }^ n$ into $A_{\mathrm{max}}$. Since every vertex of $A$ is contained in $B$, we must have $n > 0$. Moreover, it follows from $(\ast )$ together with Proposition 7.1.8.3 that the vertex $a = \sigma (0)$ is a $U'$-initial object of $\operatorname{\mathcal{C}}'$. Applying Corollary 7.1.6.13, we deduce that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r]^-{ g_{\mathrm{max}} \circ \sigma } \ar [d] & \operatorname{\mathcal{C}}' \ar [d]^{U'} \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r]^-{ g_0 \circ \sigma } & \operatorname{\mathcal{D}}' } \]

has a solution, which contradicts the maximality of $(A_{\mathrm{max}}, g_{\mathrm{max}} )$. $\square$

Corollary 7.1.8.7. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $B$ and $K$ be simplicial sets, and suppose we are given a lifting problem

7.8
\begin{equation} \begin{gathered}\label{equation:relative-colimit-pointwise-existence} \xymatrix@R =50pt@C=50pt{ B \times K \ar [r]^-{ f } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ B \times K^{\triangleright } \ar [r]^-{\overline{g}} \ar@ {-->}[ur]^{ \overline{f} } & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

Assume that, for each vertex $b \in B$, the restriction $f|_{\{ b\} \times K}$ can be extended to a $U$-colimit diagram $\overline{f}_{b}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $U \circ \overline{f}_{b} = \overline{g}|_{ \{ b\} \times K^{\triangleright } }$. Then the lifting problem (7.8) admits a solution $\overline{f}: B \times K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{f}|_{ \{ b\} \times K^{\triangleright } } = \overline{f}_{b}$ for each $b \in B$.

Proof. Apply Corollary 7.1.8.6 in the special case where $A = \operatorname{sk}_0(B)$ is the $0$-skeleton of $B$. $\square$

We can now prove a weak form of Proposition 7.1.8.1:

Corollary 7.1.8.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $f: K \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the diagram

\[ K \xrightarrow {f} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]

has a colimit in $\operatorname{\mathcal{C}}$. Then $f$ can be extended to a morphism $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ having the property that each composition $K^{\triangleright } \xrightarrow {\overline{f}} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Proof. Apply Corollary 7.1.8.7 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$. $\square$

To complete the proof of Proposition 7.1.8.1, we must show that the morphism $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ appearing in the statement of Corollary 7.1.8.8 is a colimit diagram. As above, it will be convenient to deduce this from a stronger assertion about relative colimit diagrams.

Proposition 7.1.8.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Let $B$ be a simplicial set and let $A$ be a simplicial subset, so that $F$ induces a functor

\[ F': \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}( B, \operatorname{\mathcal{D}}). \]

Suppose we are given a diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ satisfying the following condition:

$(\ast )$

Let $\sigma : \Delta ^ n \rightarrow B$ be an $n$-simplex of $B$ which is not contained in $A$ and set $b = \sigma (0)$. Then the composite map $K^{\triangleright } \xrightarrow { \overline{f} } \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}}$ is an $F$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.

Then $\overline{f}$ is an $F'$-colimit diagram in the $\infty $-category $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$.

Proof. As in the proof of Corollary 7.1.8.6, we can replace $F$ by the restriction functor

\[ \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) \]

and thereby reduce to the special case $K = \emptyset $ (Proposition 7.1.8.3). In this case, we view $\overline{f}$ as an object of the $\infty $-category $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$, and we wish to show that this object is $F'$-initial.

Using Proposition 4.1.3.2, we can factor $F$ as a composition $\operatorname{\mathcal{C}}\xrightarrow {G} \operatorname{\mathcal{E}}\xrightarrow {U} \operatorname{\mathcal{D}}$, where $U$ is an inner fibration (so that $\operatorname{\mathcal{E}}$ is an $\infty $-category) and $G$ is inner anodyne (and therefore an equivalence of $\infty $-categories). Note that we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \ar [d]^{G \circ } \ar [r]^-{F'} & \operatorname{Fun}(A,\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B,\operatorname{\mathcal{D}}) \ar [r] \ar [d] & \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \ar [d]^{G \circ } \\ \operatorname{Fun}(B, \operatorname{\mathcal{E}}) \ar [r]^-{U'} & \operatorname{Fun}(A,\operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B,\operatorname{\mathcal{D}}) \ar [r] & \operatorname{Fun}(A, \operatorname{\mathcal{E}}), } \]

where the vertical maps on the left and right are equivalences of $\infty $-categories (Remark 4.5.1.16). Since the square on the right is a pullback diagram and the right horizontal maps are isofibrations (Corollary 4.4.5.3), it follows that the vertical map in the middle is also an equivalence of $\infty $-categories (Corollary 4.5.4.6). Consequently, to show that $\overline{f}$ is $F'$-initial, it will suffice to show that $G \circ \overline{f}$ is $U'$-initial when viewed as an object of $\operatorname{Fun}(B, \operatorname{\mathcal{E}})$ (Remark 7.1.6.8). Since $U'$ is an inner fibration (Proposition 4.1.4.1), it will suffice to verify that $\overline{f}$ satisfies the criterion of Corollary 7.1.6.13: every lifting problem

7.9
\begin{equation} \begin{gathered}\label{equation:relative-colimit-pointwise-universal-strong} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{Fun}(B, \operatorname{\mathcal{E}}) \ar [d]^{ U' } \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}(A,\operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B,\operatorname{\mathcal{D}}) } \end{gathered} \end{equation}

has a solution, provided that $n > 0$ and $\sigma _0(0) = \overline{f}$. Unwinding the definitions, we can rewrite (7.9) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ (\operatorname{\partial \Delta }^ n \times B) \coprod _{(\operatorname{\partial \Delta }^ n \times A) } (\Delta ^ n \times B) \ar [r]^-{ g } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Delta ^ n \times B \ar [r] & \operatorname{\mathcal{D}}. } \]

Since $n > 0$, every vertex of the simplicial set $\Delta ^ n \times B$ is contained in $\operatorname{\partial \Delta }^ n \times B$. Moreover, if $\tau : \Delta ^ m \rightarrow \Delta ^ n \times B$ is an $m$-simplex which does not belong to $(\operatorname{\partial \Delta }^ n \times B) \coprod _{(\operatorname{\partial \Delta }^ n \times A) } (\Delta ^ n \times B)$, then condition $(\ast )$ (and Remark 7.1.6.8) guarantee that $g$ carries $\tau (0)$ to a $U'$-initial vertex of $\operatorname{\mathcal{E}}$. The existence of the desired solution now follows from Corollary 7.1.8.6 (applied in the special case $K = \emptyset $). $\square$

Corollary 7.1.8.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset, and let $U: \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A,\operatorname{\mathcal{C}})$ be the restriction functor. Let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram satisfying the following condition:

$(\ast )$

Let $\sigma : \Delta ^ n \rightarrow B$ be an $n$-simplex of $B$ which is not contained in $A$ and set $b = \sigma (0)$. Then the composite map $K^{\triangleright } \xrightarrow { \overline{f} } \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.

Then $\overline{f}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$.

Proof. Apply Proposition 7.1.8.9 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$. $\square$

Corollary 7.1.8.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $B$ be a simplicial set, and let $F': \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ be given by composition with $F$. Let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the composition

\[ K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]

is an $F$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $\overline{f}$ is an $F'$-colimit diagram in the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$.

Proof. Apply Proposition 7.1.8.9 in the special case $A = \emptyset $. $\square$

Corollary 7.1.8.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram. Assume that, for each vertex $b \in B$, the composite map $K^{\triangleright } \xrightarrow { \overline{f} } \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Then $\overline{f}$ is a colimit diagram in $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$.

Proof. Apply Corollary 7.1.8.11 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$ (or Corollary 7.1.8.10) in the special case $A = \emptyset $). $\square$

Proof of Proposition 7.1.8.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $f: K \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the composite diagram

\[ K \xrightarrow {f} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]

admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}$. Applying Corollary 7.1.8.8, we see that $f$ admits an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ with the property that, for every vertex $b \in B$, the composition $\operatorname{ev}_{b} \circ \overline{f}$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Applying Corollary 7.1.8.12, we see any such extension is a colimit diagram in $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$. To complete the proof, it will suffice to show the converse: if $\overline{f}': K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ is any colimit diagram extending $f$ and $b \in B$ is a vertex, then $\operatorname{ev}_{b} \circ \overline{f}'$ is also a colimit diagram in $\operatorname{\mathcal{C}}$. In this case, the extension $\overline{f}'$ is isomorphic to $\overline{f}$ as an object of the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{Fun}(B,\operatorname{\mathcal{C}}) )$. It follows that $\operatorname{ev}_{b} \circ \overline{f}'$ is isomorphic to $\operatorname{ev}_{b} \circ \overline{f}$ as an object of the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ and therefore a colimit diagram by virtue of Corollary 7.1.5.7. $\square$