To study the behavior of a (large) category $\operatorname{\mathcal{D}}$, it is often useful to approximate $\operatorname{\mathcal{D}}$ by well-chosen (small) subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$. The following condition guarantees that, for some purposes, passage from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$ does not lose too much information:
Definition 8.4.1.1. Let $\operatorname{\mathcal{D}}$ be a (locally small) category. We say that a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if the functor is fully faithful.
\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \bullet , Y ) \]
Remark 8.4.1.2. Definition 8.4.1.1 was introduced by Isbell in [MR175954]. Beware that Isbell uses the term left adequate subcategory for what we refer to as a dense subcategory.
Example 8.4.1.3. Let $\operatorname{Cat}$ denote the ordinary category whose objects are small categories and whose morphisms are functors, and let $\operatorname{{\bf \Delta }}\subset \operatorname{Cat}$ be the simplex category. Proposition 1.3.3.1 asserts that the restricted Yoneda embedding is fully faithful, so that $\operatorname{{\bf \Delta }}$ is a dense subcategory of $\operatorname{Cat}$.
\[ \operatorname{Cat}\rightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set}) = \operatorname{Set_{\Delta }}\quad \quad \operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \]
Exercise 8.4.1.4. Let $\operatorname{\mathcal{C}}$ denote the category of partially ordered sets, and let $\operatorname{{\bf \Delta }}_{\leq 1}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the objects $[0]$ and $[1]$. Show that $\operatorname{{\bf \Delta }}_{\leq 1}$ is a dense subcategory of $\operatorname{\mathcal{C}}$.
We now introduce an $\infty $-categorical counterpart of Definition 8.4.1.1.
Definition 8.4.1.5. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. We will say that a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if, for every object $X \in \operatorname{\mathcal{D}}$, the composition is a colimit diagram.
\[ ( \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X} )^{\triangleright } \hookrightarrow \operatorname{\mathcal{D}}_{/X}^{\triangleright } \rightarrow \operatorname{\mathcal{D}} \]
Remark 8.4.1.6. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if and only if the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$ is left Kan extended from $\operatorname{\mathcal{C}}$.
Example 8.4.1.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is a dense full subcategory of itself (see Example 7.3.3.8).
In the situation of Definition 8.4.1.5, suppose that the $\infty $-category $\operatorname{\mathcal{D}}$ is locally small, and let $h_{\bullet }: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ be a covariant Yoneda embedding for $\operatorname{\mathcal{D}}$ (see Definition 8.3.3.9). Composing with the restriction functor $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, we obtain a functor
which we will refer to as the restricted Yoneda embedding.
Proposition 8.4.1.8. Let $\operatorname{\mathcal{D}}$ be a locally small $\infty $-category. A full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if and only if the restricted Yoneda embedding $\operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is fully faithful.
We will deduce Proposition 8.4.1.8 from a more general result (Proposition 8.4.1.22), which we prove at the end of this section.
Corollary 8.4.1.9. Let $\operatorname{\mathcal{D}}$ be a locally small category. Then a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense (in the sense of Definition 8.4.1.1) if and only if $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a dense subcategory of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ (in the sense of Definition 8.4.1.5).
Warning 8.4.1.10. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Consider the following conditions on a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$:
The $\infty $-category $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense, in the sense of Definition 8.4.1.5.
Every object $X \in \operatorname{\mathcal{D}}$ can be realized as the colimit of a diagram taking values in the full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$.
The $\infty $-category $\operatorname{\mathcal{D}}$ is generated by $\operatorname{\mathcal{C}}$ under colimits. That is, if $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ is a full subcategory which contains $\operatorname{\mathcal{C}}$ and is closed under the formation of colimits in $\operatorname{\mathcal{D}}$, then $\operatorname{\mathcal{D}}_0 = \operatorname{\mathcal{D}}$.
It follows immediately from the definitions that $(1) \Rightarrow (2) \Rightarrow (3)$. Beware that neither of this implications is reversible. See Exercises 8.4.1.11 and 8.4.1.12.
Exercise 8.4.1.11. Let $\operatorname{\mathcal{D}}$ denote the category of free abelian groups, and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ denote the full subcategory spanned by object $\operatorname{\mathbf{Z}}$. Show that $\operatorname{\mathcal{C}}$ is not a dense subcategory of $\operatorname{\mathcal{D}}$. Consequently, the inclusion map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \subset \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ satisfies condition $(2)$ of Warning 8.4.1.10, but does not satisfy condition $(1)$.
Exercise 8.4.1.12. Let $\operatorname{Cat}$ denote the (ordinary) category of small categories, and let $\operatorname{{\bf \Delta }}_{\leq 1} \subset \operatorname{Cat}$ denote the full subcategory spanned by the objects $[0]$ and $[1]$. Show that:
The full subcategory $\operatorname{{\bf \Delta }}_{\leq 1}$ generates $\operatorname{Cat}$ under colimits.
A small category $\operatorname{\mathcal{C}}$ can be realized as the colimit (in $\operatorname{Cat}$) of a diagram $\mathcal{K} \rightarrow \operatorname{{\bf \Delta }}_{\leq 1}$ if and only if the category $\operatorname{\mathcal{C}}$ is free, in the sense of Definition 1.3.7.7.
In particular, the inclusion $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\leq 1} ) \subset \operatorname{N}_{\bullet }( \operatorname{Cat})$ satisfies condition $(3)$ of Warning 8.4.1.10, but does not satisfy condition $(2)$.
Warning 8.4.1.13 (Failure of Transitivity). Let $\operatorname{\mathcal{E}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{E}}$ be full subcategories. Suppose that $\operatorname{\mathcal{C}}$ is a dense subcategory of $\operatorname{\mathcal{E}}$. Then $\operatorname{\mathcal{C}}$ is also a dense subcategory of $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{D}}$ is a dense subcategory of $\operatorname{\mathcal{E}}$ (see Corollary 7.3.8.8). Beware that the converse is false (Example 8.4.1.14).
Example 8.4.1.14. Let $\operatorname{Cat}$ denote the (ordinary) category of small categories. Then the simplex category $\operatorname{{\bf \Delta }}$ is a dense full subcategory of $\operatorname{Cat}$ (Example 8.4.1.3), and $\operatorname{{\bf \Delta }}_{\leq 1}$ is a dense full subcategory of $\operatorname{{\bf \Delta }}$ (Exercise 8.4.1.4). However, $\operatorname{{\bf \Delta }}_{\leq 1}$ is not a dense full subcategory of $\operatorname{Cat}$ (Exercise 8.4.1.12).
For some applications, it will be useful to consider the following generalization of Definition 8.4.1.5.
Definition 8.4.1.15. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. We say that $F$ is dense if the identity transformation $\operatorname{id}_{F}: F \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}} \circ F$ exhibits the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as a left Kan extension of $F$ along $F$ (see Variant 7.3.1.5).
Example 8.4.1.16. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense (in the sense of Definition 8.4.1.5) if and only if the inclusion functor $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is dense (in the sense of Definition 8.4.1.15). See Proposition 7.3.2.6.
Remark 8.4.1.17 (Homotopy Invariance). Let $\operatorname{\mathcal{D}}$ be an $\infty $-category, let $C$ be a simplicial set, and let $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be diagrams which are isomorphic (when viewed as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). Then $F$ is dense if and only if $F'$ is dense. This follows by combining Remarks 7.3.1.10 and 7.3.1.11.
Remark 8.4.1.18 (Change of Source). Let $\operatorname{\mathcal{D}}$ be an $\infty $-category, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, and let $G: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$ be a categorical equivalence of simplicial sets. Then $F$ is dense if and only if $F \circ G$ is dense. See Proposition 7.3.1.14.
Remark 8.4.1.19 (Change of Target). Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then:
If $G$ is fully faithful and $G \circ F$ is dense, then $F$ is dense.
If $G$ is an equivalence of $\infty $-categories and $F$ is dense, then $G \circ F$ is dense.
See Remark 7.3.1.13.
Remark 8.4.1.20. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $f$ is dense if and only if, for every object $Y \in \operatorname{\mathcal{D}}$, the composite map is a colimit diagram in $\operatorname{\mathcal{D}}$.
\[ (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} )^{\triangleright } \rightarrow \operatorname{\mathcal{D}}_{/Y}^{\triangleright } \rightarrow \operatorname{\mathcal{D}} \]
Remark 8.4.1.21. Let $\kappa $ be an uncountable regular cardinal and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and that $\operatorname{\mathcal{D}}$ is a locally $\kappa $-small $\infty $-category. Then, for each object $X \in \operatorname{\mathcal{D}}$, the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X}$ is also essentially $\kappa $-small (Corollary 5.6.7.7). Let $\lambda $ be an infinite cardinal satisfying $\mathrm{ecf}(\lambda ) \geq \kappa $ (see Definition 4.7.3.16). Then $F$ is dense if and only if, for every representable functor $h_{Y}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, the identity transformation $\operatorname{id}: h_{Y} \circ F^{\operatorname{op}} \rightarrow h_{Y} \circ F^{\operatorname{op}}$ exhibits the functor $h_{Y}$ as a right Kan extension of $h_{Y} \circ F^{\operatorname{op}}$ along $F^{\operatorname{op}}$. This follows by combining Remark 8.4.1.20 with Proposition 7.4.5.16 (together with Remark 7.4.5.18).
Proposition 8.4.1.22. Let $\lambda $ be an uncountable cardinal, let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is locally $\lambda $-small, and let be a covariant Yoneda embedding for $\operatorname{\mathcal{D}}$ (Definition 8.3.3.9). If $\operatorname{\mathcal{C}}$ is a simplicial set, then a diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is dense if and only if the composite functor is fully faithful.
\[ h_{\bullet }: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } ) \quad \quad Y \mapsto h_{Y} \]
\[ \operatorname{\mathcal{D}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) \xrightarrow { \circ F^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) \]
Proof. Choose an uncountable cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and $\operatorname{\mathcal{D}}$ is locally $\kappa $-small. Enlarging $\lambda $ if necessary, we may assume that the exponential cofinality of $\lambda $ is $\geq \kappa $ (see Remark 4.7.3.19). For each object $Y \in \operatorname{\mathcal{D}}$, let $h_{Y}^{\circ }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ denote the composite functor $h_{Y} \circ F^{\operatorname{op}}$, given on objects by the construction $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), Y )$. By virtue of Remark 8.4.1.21, it will suffice to show that the following conditions are equivalent:
The identity transformation $\operatorname{id}: h_{Y} \circ F^{\operatorname{op}} \rightarrow h_{Y}$ exhibits $h_{Y}$ as a right Kan extension of $h_{Y}^{\circ }$ along the functor $F^{\operatorname{op}}$.
For each object $X \in \operatorname{\mathcal{C}}$, the composite map
is a homotopy equivalence of Kan complexes.
Since the covariant Yoneda embedding $X \mapsto h_{X}$ is fully faithful (Theorem 8.3.3.13), we can reformulate $(2_ Y)$ as follows:
The restriction map
is a homotopy equivalence of Kan complexes.
The inequality $\kappa \leq \mathrm{ecf}(\lambda )$ guarantees that the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ admits $\kappa $-small limits (Corollary 7.4.1.13). Using Proposition 7.6.7.13, we can choose a functor $\mathscr {G}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{<\lambda }$ and a natural transformation $\alpha : \mathscr {G} \circ F^{\operatorname{op}} \rightarrow h_{Y}^{\circ }$ which exhibits $\mathscr {G}$ as a right Kan extension of $h_{Y}^{\circ }$ along the functor $F^{\operatorname{op}}$. Invoking the universal mapping property of $\mathscr {G}$ (Proposition 7.3.6.1), we see that there exists a natural transformation $\beta : h_{Y} \rightarrow \mathscr {G}$ and a commutative diagram
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$. Using Remark 7.3.1.12, we see that condition $(1_ Y)$ is satisfied if and only if the natural transformation $\beta $ is an isomorphism: that is, it induces a homotopy equivalence of Kan complexes $\beta _{X}: h_{Y}(X) \rightarrow \mathscr {G}(X)$ for each object $X \in \operatorname{\mathcal{D}}$ (Theorem 4.4.4.4). Combining this observation with Proposition 8.3.1.3, we can reformulate $(1_ Y)$ as follows:
For each object $X \in \operatorname{\mathcal{C}}$, precomposition with $\beta $ induces a homotopy equivalence
Using the commutativity of (8.60), we see that the diagram of Kan complexes
commutes up to homotopy, where the diagonal map on the right is the homotopy equivalence of Proposition 7.3.6.1. It follows that conditions $(1'_ Y)$ and $(2'_ Y)$ are equivalent. $\square$
Proof of Proposition 8.4.1.8. Let $\operatorname{\mathcal{D}}$ be a locally small $\infty $-category and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ be a full subcategory. By virtue of Example 8.4.1.16, it will suffice to show that the inclusion functor $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is dense if and only if the restricted Yoneda embedding $\operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is fully faithful. Following the convention of Remark 4.7.0.5, this is a special case of Proposition 8.4.1.22. $\square$
Proposition 8.4.1.23. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, and let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be a full subcategory which contains the image of $F$. If $F$ is dense, then the subcategory $\operatorname{\mathcal{D}}_0$ is dense.
Proof. Using Proposition 4.1.3.2, we can factor $F$ as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{C}}' \xrightarrow {F''} \operatorname{\mathcal{D}}$, where $F'$ is inner anodyne and $F''$ is an inner fibration. Using Remark 8.4.1.18, we see that the functor $F'$ is dense. Replacing $F$ by $F''$, we can reduce to proving Proposition 8.4.1.23 in the special case where $\operatorname{\mathcal{C}}$ is an $\infty $-category.
Let $\gamma $ denote the identity map $\operatorname{id}_{F}$, which we regard as a natural transformation from $F$ to $\operatorname{id}_{\operatorname{\mathcal{D}}} \circ F$. Our assumption that $F$ is dense guarantees that $\gamma $ exhibits $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as a left Kan extension of $F$ along $F$. To avoid confusion, let us write $F_0$ to denote the functor $F$, regarded as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}_0$. Let $\iota : \operatorname{\mathcal{D}}_0 \hookrightarrow \operatorname{\mathcal{D}}$ denote the inclusion map, so that $F = \iota \circ F_0$. We can therefore also regard the identity map $\operatorname{id}_{F}$ as a natural transformation $\alpha : F \rightarrow \iota \circ F_0$. Our assumption that $F$ is dense also guarantees that $\alpha $ exhibits $\iota $ as a left Kan extension of $F$ along $F_0$. Note that $\gamma = \operatorname{id}_{F}$ is a composition of $\alpha = \operatorname{id}_{F}$ with $\beta |_{ \operatorname{\mathcal{C}}}$, where $\beta = \operatorname{id}_{ \iota }$ is the identity transformation from $\iota $ to itself. Invoking the transitivity of Kan extensions (Proposition 7.3.8.18), we deduce that $\beta $ exhibits the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as a left Kan extension of $\iota $ along itself: that is, the functor $\iota $ is dense. Applying Example 8.4.1.16, we conclude that $\operatorname{\mathcal{D}}_0$ is a dense subcategory of $\operatorname{\mathcal{D}}$. $\square$
Remark 8.4.1.24. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a dense diagram. Choose another diagram $q: K \rightarrow \operatorname{\mathcal{D}}$ and set $\widetilde{\operatorname{\mathcal{D}}} = \operatorname{\mathcal{D}}_{/q}$, so that the projection map $\pi : \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ is a right fibration. Then the projection map $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}} \rightarrow \widetilde{\operatorname{\mathcal{D}}}$ is dense. To prove this, it will suffice to show that for every object $Y \in \widetilde{\operatorname{\mathcal{D}}}$, the induced map is a colimit diagram in $\widetilde{\operatorname{\mathcal{D}}}$ (Remark 8.4.1.20). By virtue of Proposition 7.1.3.19, this is equivalent to the requirement that $\pi \circ \theta $ is a colimit diagram in $\operatorname{\mathcal{D}}$. Unwinding the definitions, we see that $\pi \circ \theta $ is given by the composition Since $\pi $ is a right fibration, the map $\widetilde{\operatorname{\mathcal{D}}}_{/Y} \rightarrow \operatorname{\mathcal{D}}_{ / \pi (Y) }$ is a trivial Kan fibration (Proposition 4.3.7.12). Using Corollary 7.2.2.2, we are reduced to showing that the map $(\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / \pi (Y) })^{\triangleright } \rightarrow \operatorname{\mathcal{D}}_{ / \pi (Y) }^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram, which follows from our assumption that $F$ is dense (Remark 8.4.1.20).
\[ \theta : (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}_{/Y} )^{\triangleright } \rightarrow \widetilde{\operatorname{\mathcal{D}}}_{/Y}^{\triangleright } \rightarrow \widetilde{\operatorname{\mathcal{D}}} \]
\[ (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}_{/Y} )^{\triangleright } \rightarrow (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / \pi (Y) })^{\triangleright } \rightarrow \operatorname{\mathcal{D}}_{ / \pi (Y) }^{\triangleright } \rightarrow \operatorname{\mathcal{D}}. \]