4.6.4 Oriented Fiber Products
Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be categories. To every pair of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, one can associate the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, whose objects are triples $(C, D, \eta )$ where $C$ is an object of $\operatorname{\mathcal{C}}$, $D$ is an object of $\operatorname{\mathcal{D}}$, and $\eta : F(C) \rightarrow G(D)$ is a morphism in the category $\operatorname{\mathcal{E}}$ (Notation 2.1.4.19). This construction has a counterpart in the setting of $\infty $-categories.
Definition 4.6.4.1 (The Oriented Fiber Product). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. We let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the simplicial set given by the iterated fiber product
\[ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{E}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}(\{ 1\} , \operatorname{\mathcal{E}}) } \operatorname{\mathcal{D}}. \]
We will refer to $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ as the oriented fiber product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ over $\operatorname{\mathcal{E}}$.
As our notation suggests, we will be primarily interested in the special case of Definition 4.6.4.1 where the simplicial sets $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are $\infty $-categories.
Proposition 4.6.4.2. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category, and suppose we are given morphisms of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Then the projection map $\theta : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration of simplicial sets.
Proof.
By construction, we have a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [r] \ar [d]^{\theta } & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \ar [d]^{\theta _0} \\ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{E}}). } \]
Since $\operatorname{\mathcal{E}}$ is an $\infty $-category, the restriction map $\theta _0$ is an isofibration of $\infty $-categories (Corollary 4.4.5.3). Invoking Remark 4.5.5.11, we conclude that $\theta $ is an isofibration of simplicial sets.
$\square$
Corollary 4.6.4.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Then the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty $-category.
Proof.
By virtue of Proposition 4.6.4.2, the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration. Since $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an $\infty $-category, it follows that $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty $-category (Remark 4.5.5.7).
$\square$
Example 4.6.4.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors between ordinary categories, and let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the oriented fiber product of Notation 2.1.4.19. Since the nerve construction is compatible with the formation of inverse limits and functor categories, we have a canonical isomorphism of simplicial sets
\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}) \simeq ( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})} \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ). \]
Consequently, Definition 4.6.4.1 can be viewed as a generalization of the classical oriented fiber product.
Example 4.6.4.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets. Then the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\Delta ^0} \operatorname{\mathcal{D}}$ can be identified with the cartesian product $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$.
Example 4.6.4.10. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, which we identify with morphisms of simplicial sets $X,Y: \Delta ^{0} \rightarrow \operatorname{\mathcal{C}}$. Then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 4.6.1.1 is the oriented fiber product $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $.
The following result is a relative version of Proposition 4.6.1.10:
Proposition 4.6.4.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. Then the projection map $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration and the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.
Proof.
We will prove the second assertion; the first follows by a similar argument. Let $A \hookrightarrow B$ be a right anodyne morphism of simplicial sets; we wish to show that every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}} \]
admits a solution. Unwinding the definitions, we are reduced to showing that a map of simplicial sets
\[ \sigma _0: B \coprod _{A} (A \diamond \{ X\} ) \rightarrow \operatorname{\mathcal{C}} \]
can be extended to a map $\sigma : B \diamond \{ X\} \rightarrow \operatorname{\mathcal{C}}$ (see Notation 4.5.8.3). By virtue of Lemma 4.5.5.2, it will suffice to show that the inclusion map
\[ \iota : B \coprod _{A} ( A \diamond \{ X\} ) \hookrightarrow B \diamond \{ X\} \]
is a categorical equivalence of simplicial sets, which follows from Corollary 4.5.8.14.
$\square$
Corollary 4.6.4.12. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category containing an object $X$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then the projection map $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration, and the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.
Proof.
Unwinding the definition, we have pullback diagrams
\[ \xymatrix@R =50pt@C=50pt{ \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\ar [r] \ar [d] & \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [d] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \ar [r] \ar [d] & \operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}& \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}. } \]
The desired result now follows by combining Proposition 4.6.4.11 with Remark 4.2.1.8.
$\square$
If $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is a functor between ordinary categories, then Remark 4.3.1.11 supplies canonical isomorphisms
\[ \operatorname{\mathcal{C}}_{/F} \simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \{ F\} \quad \quad \operatorname{\mathcal{C}}_{F/} \simeq \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]
Our next goal is to establish a similar result in the setting of $\infty $-categories. Here the situation is a bit more subtle: if $F: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram in an $\infty $-category $\operatorname{\mathcal{C}}$, then the simplicial sets $\operatorname{\mathcal{C}}_{/F}$ and $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $ are generally not isomorphic. However, we will show that they are equivalent as $\infty $-categories.
Construction 4.6.4.13 (The Slice Diagonal Morphism). Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $c: \operatorname{\mathcal{C}}_{/F} \diamond K \rightarrow \operatorname{\mathcal{C}}_{/F} \star K$ be the comparison morphism of Notation 4.5.8.3. By virtue of Remark 4.6.4.9, the composite map
\[ \operatorname{\mathcal{C}}_{/F} \diamond K \xrightarrow {c} \operatorname{\mathcal{C}}_{/F} \star K \rightarrow \operatorname{\mathcal{C}} \]
determines a morphism of simplicial sets $\delta _{/F}: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $, which we will refer to as the slice diagonal morphism. Similarly, the composition
\[ K \diamond \operatorname{\mathcal{C}}_{F/} \rightarrow K \star \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{C}} \]
determines a morphism of simplicial sets $\delta _{F/}: \operatorname{\mathcal{C}}_{F/} \rightarrow \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$, which we will refer as the coslice diagonal morphism.
Exercise 4.6.4.16. Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then $f$ can be identified with a vertex of the simplicial set $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, which (to avoid confusion) we will temporarily denote by $F$. Applying Construction 4.6.4.13 to the inclusion map $\{ F \} \hookrightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$, we obtain a monomorphism of simplicial sets $\operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \hookrightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ F\} $, which induces a monomorphism
\[ u: \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ F\} . \]
Show that the slice diagonal morphism $\delta _{/f}: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $ of Construction 4.6.4.13 factors (uniquely) through $u$. In particular, $\delta _{/f}$ determines a morphism of simplicial sets $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F}$. Similarly, the coslice diagonal morphism $\delta _{f/}$ induces a morphism of simplicial sets $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{F/} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$.
We can now formulate our main result, which we prove at the end of this section.
Theorem 4.6.4.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the slice and coslice diagonal maps
\[ \delta _{/F}: \operatorname{\mathcal{C}}_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \quad \quad \delta _{F/}: \operatorname{\mathcal{C}}_{F/} \hookrightarrow \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}} \]
are equivalences of $\infty $-categories.
Corollary 4.6.4.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C \in \operatorname{\mathcal{C}}$ be an object. Then the slice and coslice diagonal maps
\[ \delta _{/C}: \operatorname{\mathcal{C}}_{/C} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} \quad \quad \delta _{C/}: \operatorname{\mathcal{C}}_{C/} \hookrightarrow \{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}} \]
are equivalences of $\infty $-categories.
Corollary 4.6.4.19. Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$. Then the induced functors
\[ G': \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{D}}_{/(G \circ F)} \quad \quad G'': \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{D}}_{(G \circ F)/} \]
are equivalences of $\infty $-categories.
Proof.
We will show that $G'$ is an equivalence of $\infty $-categories; the analogous statement for $G''$ follows by a similar argument. Note that we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/F} \ar [r]^-{G'} \ar [d] & \operatorname{\mathcal{D}}_{ / (G \circ F) } \ar [d] \\ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \ar [r]^-{\overline{G}'} & \operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \{ G \circ F \} , } \]
where the vertical maps are equivalences of $\infty $-categories by virtue of Theorem 4.6.4.17. It will therefore suffice to show that $\overline{G}'$ is an equivalence of $\infty $-categories, which is a special case of Remark 4.6.4.4.
$\square$
Corollary 4.6.4.20. Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be fully faithful functor of $\infty $-categories and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the induced functors
\[ G': \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{D}}_{/(G \circ F)} \quad \quad G'': \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{D}}_{(G \circ F)/} \]
are also fully faithful.
Proof.
Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{D}}$ be the essential image of $G$ (Definition 4.6.2.12), so that $G$ induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ (Corollary 4.6.2.23). By virtue of Corollary 4.6.4.19, the functors $G'$ and $G''$ restrict to equivalences
\[ \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}'_{/(G \circ F)} \quad \quad \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{C}}'_{(G \circ F)/} \]
We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$ and thereby reduce to the case where $G: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is the inclusion of a full subcategory. In this case, the functors $G'$ and $G''$ are also the inclusions of full subcategories, hence fully faithful (Example 4.6.2.2).
$\square$
We now turn to the proof of Theorem 4.6.4.17. As we will see, it is essentially a reformulation of Theorem 4.5.8.8.
Lemma 4.6.4.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram indexed by a simplicial set $K$. Suppose we are given a pair of diagrams $e_0, e_1: J \rightarrow \operatorname{\mathcal{C}}_{/F}$ indexed by a simplicial set $J$, which we identify with diagrams $F_0, F_1: J \star K \rightarrow \operatorname{\mathcal{C}}$ satisfying $F_0|_{K} = F = F_{1}|_{K}$. The following conditions are equivalent:
- $(1)$
The diagrams $e_0$ and $e_1$ are isomorphic when regarded as objects of the diagram $\infty $-category $\operatorname{Fun}(J, \operatorname{\mathcal{C}}_{/F} )$.
- $(2)$
The diagrams $F_0$ and $F_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{ K / }(J \star K, \operatorname{\mathcal{C}})$.
Proof.
Choose a categorical mapping cylinder
\[ J \coprod J \xrightarrow {(s_0, s_1)} \overline{J} \xrightarrow {\pi } J \]
for the simplicial set $J$ (Definition 4.6.3.3). Using Corollary 4.5.8.9, we deduce that the resulting diagram
\[ (J \star K) \coprod _{K} (J \star K) \xrightarrow {(s'_0, s'_1)} \overline{J} \star K \xrightarrow {\pi '} J \star K \]
is a categorical mapping cylinder for the join $J \star K$ relative to $K$. Using the criterion of Corollary 4.6.3.11, we see that $(1)$ and $(2)$ can be reformulated as follows:
- $(1')$
There exists a diagram $\overline{e}: \overline{J} \rightarrow \operatorname{\mathcal{C}}_{/F}$ satisfying $e_0 = \overline{e} \circ s_0$ and $e_1 = \overline{e} \circ s_1$.
- $(2')$
There exists a diagram $\overline{F}': \overline{J} \star K \rightarrow \operatorname{\mathcal{C}}$ satisfying $F_0 = \overline{F} \circ s'_0$ and $F_1 = \overline{F} \circ s'_1$.
The equivalence of $(1')$ and $(2')$ follows immediately from the universal property of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/F}$.
$\square$
Proof of Theorem 4.6.4.17.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, which we regard as an object of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$. We will show that the slice diagonal morphism
\[ \delta _{/F}: \operatorname{\mathcal{C}}_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \]
is an equivalence of $\infty $-categories; the corresponding assertion for the coslice diagonal morphism follows by a similar argument. Fix a simplicial set $J$; we wish to show that the induced map of sets
\[ \theta : \pi _0( \operatorname{Fun}(J, \operatorname{\mathcal{C}}_{/F} )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( J, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} )^{\simeq } ) \]
is a bijection. Using Lemma 4.6.4.21 and Remark 4.6.4.14, we can identify $\theta $ with the map of sets
\[ \pi _0( \operatorname{Fun}_{K/}( J \star K, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}_{K/}( J \diamond K, \operatorname{\mathcal{C}})^{\simeq } ) \]
induced by precomposition with the comparison map $c_{J,K}: J \diamond K \twoheadrightarrow J \star K$ of Notation 4.5.8.3. It will therefore suffice to show that composition with $c_{J,K}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{K/}( J \star K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{K/}( J \diamond K, \operatorname{\mathcal{C}})$. This follows by applying Corollary 4.5.2.32 to the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( J \star K, \operatorname{\mathcal{C}}) \ar [r]^-{\circ c_{J,K}} \ar [d] & \operatorname{Fun}( J \diamond K, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( K, \operatorname{\mathcal{C}}) \ar@ {=}[r] & \operatorname{Fun}( K, \operatorname{\mathcal{C}}); } \]
here the vertical maps are isofibrations (Corollary 4.4.5.3) and the upper horizontal map is an equivalence of $\infty $-categories because the morphism $c_{J,K}$ is a categorical equivalence (Theorem 4.5.8.8).
$\square$