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7.2.1 Cofinal Morphisms of Simplicial Sets

Recall that a morphism of simplicial sets $f: A \rightarrow B$ is left anodyne if, for every left fibration $q: X \rightarrow S$, every lifting problem

$\xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{q} \\ B \ar@ {-->}[ur] \ar [r] & S }$

admits a solution (Proposition 4.2.4.5). Beware that this condition can only be satisfied if $f$ is a monomorphism of simplicial sets, and is therefore not invariant under categorical equivalence. Our goal in this section is to introduce an enlargement of the collection of left anodyne morphisms which does not suffer from this defect.

Definition 7.2.1.1 (Joyal). Let $f: A \rightarrow B$ be a morphism of simplicial sets. We say that $f$ is left cofinal if, for every left fibration $q: \widetilde{B} \rightarrow B$, precomposition with $f$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}_{/B}( B, \widetilde{B} ) \rightarrow \operatorname{Fun}_{/B}(A, \widetilde{B} )$ (see Corollary 4.4.2.5). We say that $f$ is right cofinal if, for every right fibration $q: \widetilde{B} \rightarrow B$, precomposition with $f$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}_{/B}( B, \widetilde{B} ) \rightarrow \operatorname{Fun}_{/B}(A, \widetilde{B} )$.

Remark 7.2.1.2. Let $f: A \rightarrow B$ be a morphism of simplicial sets. Then $f$ is left cofinal if and only if the opposite morphism $f^{\operatorname{op}}: A^{\operatorname{op}} \rightarrow B^{\operatorname{op}}$ is right cofinal.

Proposition 7.2.1.3. Let $f: A \rightarrow B$ be a morphism of simplicial sets. Then $f$ is left anodyne if and only if it is a left cofinal monomorphism. Similarly, $f$ is right anodyne if and only if it is a right cofinal monomorphism.

Proof. We will prove the first assertion; the second follows by a similar argument. Assume first that $f$ is left anodyne. Then $f$ is a monomorphism (Remark 4.2.4.4). For every left fibration of simplicial sets $\widetilde{B} \rightarrow B$, the restriction map $\theta : \operatorname{Fun}_{/B}( B, \widetilde{B} ) \rightarrow \operatorname{Fun}_{/B}(A, \widetilde{B} )$ is a pullback of the map

$\operatorname{Fun}(B, \widetilde{B} ) \rightarrow \operatorname{Fun}(B,B) \times _{ \operatorname{Fun}(A,B) } \operatorname{Fun}(A, \widetilde{B} ),$

and is therefore a trivial Kan fibration (Proposition 4.2.5.4). In particular, $u$ is a homotopy equivalence (Proposition 3.1.6.10). Allowing $\widetilde{B}$ to vary, we conclude that $f$ is left cofinal.

We now prove the converse. Assume that $f$ is a left cofinal monomorphism; we wish to show that $f$ is left anodyne. By virtue of Proposition 4.2.4.5, it will suffice to show that for every lifting problem

7.7
$$\begin{gathered}\label{equation:cofinal-vs-anodyne} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{q} \\ B \ar@ {-->}[ur] \ar [r]^-{g} & S } \end{gathered}$$

admits a solution, provided that $q$ is a left fibration of simplicial sets. Let us regard the morphism $g$ as fixed, and consider the restriction map

$\theta : \operatorname{Fun}_{/B}( B, X \times _{S} B ) \rightarrow \operatorname{Fun}_{ / B }( A, X \times _{S} B).$

Since $f$ is a monomorphism, the morphism $\theta$ is a left fibration (Proposition 4.2.5.1). Since the target simplicial set $\operatorname{Fun}_{ / B }( A, X \times _{S} B)$ is a Kan complex (Corollary 4.4.2.5), it follows that $\theta$ is a Kan fibration (Corollary 4.4.3.8). Our assumption that $f$ is left cofinal guarantees that $\theta$ is a homotopy equivalence, and therefore a trivial Kan fibration (Proposition 3.2.6.15). In particular, it is surjective at the level of vertices, which guarantees that (7.7) admits a solution. $\square$

Example 7.2.1.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then the inclusion map $\{ X\} \hookrightarrow \operatorname{\mathcal{C}}$ is right cofinal if and only if $X$ is a final object of $\operatorname{\mathcal{C}}$. This follows by combining Proposition 7.2.1.3 with Corollary 4.6.6.25. Similarly, the inclusion map $\{ X \} \hookrightarrow \operatorname{\mathcal{C}}$ is left cofinal if and only if $X$ is an initial object of $\operatorname{\mathcal{C}}$.

Proposition 7.2.1.5. Let $f: A \rightarrow B$ be a morphism of simplicial sets. Then:

$(1)$

If $f$ is either left cofinal or right cofinal, then it is a weak homotopy equivalence.

$(2)$

If $f$ is a weak homotopy equivalence and $B$ is a Kan complex, then $f$ is left and right cofinal.

Proof. We first prove $(1)$. Let $X$ be a Kan complex. Then the projection map $X \times B \rightarrow B$ is a Kan fibration (Remark 3.1.1.6), and therefore both a left and a right fibration (Example 4.2.1.5). Consequently, if $f$ is either left cofinal or right cofinal, the induced map

$\operatorname{Fun}(B,X) \simeq \operatorname{Fun}_{/B}(B, X \times B) \rightarrow \operatorname{Fun}_{/B}(A, X \times B) \simeq \operatorname{Fun}(A,X)$

is a homotopy equivalence of Kan complexes. Allowing $X$ to vary, we conclude that $f$ is a weak homotopy equivalence.

We now prove $(2)$. Assume that $B$ is a Kan complex and that $f$ is a weak homotopy equivalence; we will show that $f$ is left cofinal (the proof that $f$ is right cofinal is similar). Let $q: \widetilde{B} \rightarrow B$ be a left fibration. Since $B$ is a Kan complex, $q$ is a Kan fibration (Corollary 4.4.3.8); in particular, $\widetilde{B}$ is a Kan complex. Applying Corollary 3.1.3.4, we obtain a commutative diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B, \widetilde{B} ) \ar [d]^{q \circ } \ar [r]^-{\circ f} & \operatorname{Fun}(A, \widetilde{B}) \ar [d]^{q \circ } \\ \operatorname{Fun}(B, B) \ar [r]^-{ \circ f} & \operatorname{Fun}(A,B ), }$

where the vertical maps are Kan fibrations (Corollary 3.1.3.2). Our assumption that $f$ is a weak homotopy equivalences guarantees that the horizontal maps are homotopy equivalences (Proposition 3.1.6.17). Applying Proposition 3.2.8.1, we deduce that the map $\operatorname{Fun}_{/B}( B, \widetilde{B} ) \rightarrow \operatorname{Fun}_{/B}(A, \widetilde{B})$ is also a homotopy equivalence. $\square$

Proposition 7.2.1.6. Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be morphisms of simplicial sets, and suppose that $f$ is left cofinal. Then $g$ is left cofinal if and only if the composite map $g \circ f$ is left cofinal. In particular, the collection of left cofinal morphisms is closed under composition.

Proof. Let $q: \widetilde{C} \rightarrow C$ be a left fibration of simplicial sets, and let

$\operatorname{Fun}_{/C}(C, \widetilde{C}) \xrightarrow { g^{\ast } } \operatorname{Fun}_{/C}( B, \widetilde{C} ) \xrightarrow { f^{\ast } } \operatorname{Fun}_{/C}( A, \widetilde{C} )$

be the morphisms given by precomposition with $g$ and $f$. Our assumption that $f$ is left cofinal guarantees that $f^{\ast }$ is a homotopy equivalence. It follows that $g^{\ast }$ is a homotopy equivalence if and only if $f^{\ast } \circ g^{\ast }$ is a homotopy equivalence (Remark 3.1.6.7). $\square$

Corollary 7.2.1.7. Let $f: A \hookrightarrow B$ and $g: B \hookrightarrow C$ be monomorphisms of simplicial sets. If both $f$ and $g \circ f$ are left anodyne, then $g$ is left anodyne. If $f$ and $g \circ f$ are right anodyne, then $g$ is right anodyne.

Warning 7.2.1.8. Let $g: \Delta ^1 \rightarrow \Delta ^0$ be the projection map and let $f: \{ 1\} \hookrightarrow \Delta ^1$ be the inclusion. Then $g$ and $g \circ f$ are left cofinal (Proposition 7.2.1.5). However, the morphism $f$ is not left cofinal, since it is not left anodyne (see Example 4.2.4.7). Consequently, the collection of left cofinal morphisms does not satisfy the two-out-of-three property.

Proposition 7.2.1.9. Let $A$ be a simplicial set, let $W$ be a collection of edges of $A$, and let $f: A \rightarrow B$ be a morphism of simplicial sets which exhibits $B$ as a localization of $A$ with respect to $W$ (see Definition 6.3.1.9). Then $F$ is both left and right cofinal.

Proof. We will show that $f$ is left cofinal; the proof that $f$ is right cofinal is similar. Let $q: \widetilde{B} \rightarrow B$ be a left fibration; we wish to show that composition with $f$ induces a homotopy equivalence $f^{\ast }: \operatorname{Fun}_{/B}(B, \widetilde{B} ) \rightarrow \operatorname{Fun}_{/B}(A, \widetilde{B})$. Applying Corollary 5.7.7.3 (and Remark 5.7.7.4), we deduce that there exists a pullback diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \widetilde{B} \ar [r] \ar [d]^{q} & \widetilde{\operatorname{\mathcal{C}}} \ar [d]^{Q} \\ B \ar [r]^-{g} & \operatorname{\mathcal{C}}, }$

where $Q$ is a left fibration of $\infty$-categories. Let $\operatorname{Fun}( A[W^{-1}], \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(A,\operatorname{\mathcal{C}})$ spanned by those diagrams which carry each edge of $W$ to an isomorphism in $\operatorname{\mathcal{C}}$ (Notation 6.3.1.1), and define $\operatorname{Fun}( A[W^{-1}], \widetilde{\operatorname{\mathcal{C}}} )$ similarly. We have a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( B, \widetilde{\operatorname{\mathcal{C}}} ) \ar [d]^{Q \circ } \ar [r]^-{\circ f} & \operatorname{Fun}(A[W^{-1}], \widetilde{\operatorname{\mathcal{C}}} ) \ar [r] \ar [d]^{Q \circ } & \operatorname{Fun}(A, \widetilde{\operatorname{\mathcal{C}}} ) \ar [d]^{Q \circ } \\ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \ar [r]^-{\circ f} & \operatorname{Fun}(A[W^{-1}], \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}(A, \operatorname{\mathcal{C}}), }$

where the vertical maps on both sides are left fibrations (Corollary 4.2.5.2). Since $Q$ is a left fibration of $\infty$-categories, it is conservative (Proposition 4.4.2.11), so the right side of the diagram is a pullback square. In particular, the vertical map in the middle is also a left fibration. Our assumption that $f$ exhibits $B$ as a localization fo $A$ with respect to $W$ guarantees that the left horizontal maps are equivalences of $\infty$-categories. Applying Corollary 4.5.2.26, we conclude that the map of fibers

\begin{eqnarray*} \operatorname{Fun}_{/B}(B, \widetilde{B} ) \simeq \{ g\} \times _{ \operatorname{Fun}(B,\operatorname{\mathcal{C}})} \operatorname{Fun}(B, \widetilde{\operatorname{\mathcal{C}}} ) & \rightarrow & \{ g \circ f \} \times _{ \operatorname{Fun}(A[W^{-1}], \operatorname{\mathcal{C}}) } \operatorname{Fun}(A[W^{-1}], \widetilde{\operatorname{\mathcal{C}}} ) \\ & = & \{ g \circ f \} \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \operatorname{Fun}(A, \widetilde{\operatorname{\mathcal{C}}} ) \\ & \simeq & \operatorname{Fun}_{/B}(A, \widetilde{B} ) \end{eqnarray*}

is an equivalence of $\infty$-categories, and therefore a homotopy equivalence of Kan complexes (Example 4.5.1.13). $\square$

Corollary 7.2.1.10. Let $f: A \twoheadrightarrow B$ be a universally localizing morphism of simplicial sets (see Definition 6.3.6.1). Then $f$ is both left and right cofinal.

Corollary 7.2.1.11. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then there exists a partially ordered set $(A, \leq )$ and a morphism of simplicial sets $F: \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$ which is both left and right cofinal. Moreover, if the simplicial set $\operatorname{\mathcal{C}}$ is finite, then we can arrange that the partially ordered set $(A, \leq )$ is finite.

Corollary 7.2.1.12. Let $f: A \rightarrow B$ be a categorical equivalence of simplicial sets. Then $f$ is left cofinal and right cofinal.

Corollary 7.2.1.13. Let $q: X \rightarrow S$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $q$ is left cofinal and a left fibration.

$(2)$

The morphism $q$ is right cofinal and a right fibration.

$(3)$

The morphism $q$ is a trivial Kan fibration.

Proof. If $q$ is a trivial Kan fibration, then it is both a left fibration and a right fibration (Example 4.2.1.5). Moreover, $q$ is also a categorical equivlaence of simplicial sets (Proposition 4.5.3.11), hence left and right cofinal by virtue of Corollary 7.2.1.12. This proves the implications $(3) \Rightarrow (1)$ and $(3) \Rightarrow (2)$.

We will complete the proof by showing that $(1) \Rightarrow (3)$ (the proof of the implication $(2) \Rightarrow (3)$ is similar). Assume that $q$ is a left cofinal left fibration. Then composition with $q$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}_{/S}( S, X) \rightarrow \operatorname{Fun}_{/S}(X,X)$. In particular, the morphism $q$ admits a section $f: S \rightarrow X$ such that $\operatorname{id}_{X}$ and $q \circ f$ belong to the same connected component of $\operatorname{Fun}_{/S}(X,X)$. For each vertex $s \in S$, let $X_{s} = \{ s\} \times _{S} X$ be the fiber of $q$ over $s$. Then the identity map $\operatorname{id}: X_{s} \rightarrow X_{s}$ is homotopic to the constant map $X_{s} \rightarrow \{ f(s) \} \hookrightarrow X_{s}$. It follows that the Kan complex $X_{s}$ is contractible. Allowing $s$ to vary, we conclude that the left fibration $q$ is a trivial Kan fibration (Proposition 4.4.2.14). $\square$

Corollary 7.2.1.14. Let $f: X \rightarrow Z$ be a morphism of simplicial sets. Then $f$ is left cofinal if and only if it factors as a composition $X \xrightarrow {f'} Y \xrightarrow {f''} Z$, where $f'$ is left anodyne and $f''$ is a trivial Kan fibration.

Proof. Suppose first that we can write $f = f'' \circ f'$, where $f'$ is left anodyne and $f''$ is a trivial Kan fibration. Proposition 7.2.1.3 guarantees that $f'$ is left cofinal, and Proposition 7.2.1.5 guarantees that $f''$ is left cofinal. Applying Proposition 7.2.1.6, we conclude that $f$ is also left cofinal.

We now prove the converse. Assume that $f: X \rightarrow Z$ is left cofinal. Applying Proposition 4.2.4.8, we can write $f$ as a composition $X \xrightarrow {f'} Y \xrightarrow {f''} Z$, where $f'$ is left anodyne and $f''$ is a left fibration. Then $f'$ is also left cofinal (Proposition 7.2.1.3). Applying Proposition 7.2.1.6, we deduce that $f''$ is left cofinal. It then follows from Corollary 7.2.1.13 that $f''$ is a trivial Kan fibration. $\square$

Corollary 7.2.1.15. Suppose we are given a categorical pushout diagram of simplicial sets

7.8
$$\begin{gathered}\label{equation:categorical-pushout-cofinal} \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d] & Z \ar [d] \\ X' \ar [r]^-{f'} & Z'. } \end{gathered}$$

If $f$ is left cofinal, then $f'$ is also left cofinal.

Proof. By virtue of Corollary 7.2.1.14, we may assume that $f$ factors as a composition $X \stackrel{g}Y \xrightarrow {h} Z$, where $g$ is left anodyne and $h$ is a trivial Kan fibration. Setting $Y' = Y \coprod _{X} X'$, we can expand (7.8) to a commutative diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [r]^-{g} \ar [d] & Y \ar [r]^-{h} \ar [d] & Z \ar [d] \\ X' \ar [r]^-{g'} & Y' \ar [r]^-{h'} & Z'. }$

Note that the square on the left is a pushout diagram in which the horizontal maps are monomorphisms, and therefore a categorical pushout diagram (Example 4.5.4.12). Applying Proposition 4.5.4.8, we deduce that the square on the right is also a categorical pushout diagram. Since $h$ is a categorical equivalence (Proposition 4.5.3.11), it follows that $h'$ is also a categorical equivalence (Proposition 4.5.4.10). In particular, $h'$ is left cofinal (Corollary 7.2.1.12). The morphism $g'$ is left anodyne (since it is a pushout of $g$), and is therefore also left cofinal (Proposition 7.2.1.3). Applying Proposition 7.2.1.6, we deduce that $f' = h' \circ g'$ is also left cofinal. $\square$

Corollary 7.2.1.16. The collection of left cofinal morphisms of simplicial sets is closed under the formation of filtered colimits (when regarded as a full subcategory of the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$).

Proof. For every morphism of simplicial sets $f: X \rightarrow Z$, let $X \xrightarrow {f'} Q(f) \xrightarrow {f''} Y$ be the factorization of Proposition 4.2.4.8, so that $f'$ is left anodyne, $f''$ is a left fibration, and the construction $f \mapsto Q(f)$ is a functor which commutes with filtered colimits. Using Propositions 7.2.1.5, 7.2.1.6, and Corollary 7.2.1.13, we see that $f$ is left cofinal if and only if the morphism $f'': Q(f) \rightarrow Z$ is a trivial Kan fibration. Since the collection of trivial Kan fibrations is closed under filtered colimits (Remark 1.4.5.3), it follows that the collection of left cofinal morphisms is also closed under filtered colimits. $\square$

Corollary 7.2.1.17. The collection of left anodyne morphisms of simplicial sets is closed under the formation of filtered colimits (when regarded as a full subcategory of the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$).

Corollary 7.2.1.18. Let $f: X \rightarrow Z$ be a left cofinal morphism of simplicial sets. Then, for every simplicial set $K$, the product map $(f \times \operatorname{id}_ K): X \times K \rightarrow Z \times K$ is left cofinal.

Proof. By virtue of Corollary 7.2.1.14, the morphism $f$ factors as a composition $X \xrightarrow {f'} Y \xrightarrow {f''} Z$, where $f'$ is left anodyne and $f''$ is a trivial Kan fibration. It follows that $f \times \operatorname{id}_{K}$ factors as a composition

$X \times K \xrightarrow {f' \times \operatorname{id}_ K} Y \times K \xrightarrow {f'' \times \operatorname{id}_ K } Z \times K.$

We now note that $f' \times \operatorname{id}_{K}$ is left anodyne (Proposition 4.2.5.3) and $f'' \times \operatorname{id}_{K}$ is a trivial Kan fibration (Remark 1.4.5.2). Applying Corollary 7.2.1.14, we deduce that $f \times \operatorname{id}_{K}$ is left cofinal. $\square$

Corollary 7.2.1.19. Let $f: X \rightarrow Y$ and $f': X' \rightarrow Y'$ be left cofinal morphisms of simplicial sets. Then the product map $(f \times f'): X \times X' \rightarrow Y \times Y'$ is left cofinal.

Proof. Factoring $f \times f'$ as a composition

$X \times X' \xrightarrow { f \times \operatorname{id}_{X'} } Y \times X' \xrightarrow { \operatorname{id}_{Y} \times f'} Y \times Y',$

the desired result follows by combining Corollary 7.2.1.18 with Proposition 7.2.1.6. $\square$

We now prove that cofinality is invariant under categorical equivalence.

Proposition 7.2.1.20. Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be morphisms of simplicial sets, and suppose that $g$ is a categorical equivalence. Then $f$ is left cofinal if and only if $g \circ f$ is left cofinal.

Proof. Since $g$ is a categorical equivalence, the construction $\widetilde{C} \mapsto B \times _{C} \widetilde{C}$ induces a bijection from equivalence classes of left fibrations over $C$ to equivalence classes of left fibrations over $B$ (Corollary 5.7.0.6). It follows that $f$ is left cofinal if and only if it satisfies the following condition:

$(\ast )$

For every left fibration $q: \widetilde{C} \rightarrow C$, the restriction map $f^{\ast }: \operatorname{Fun}_{/C}(B, \widetilde{C} ) \rightarrow \operatorname{Fun}_{/C}( A, \widetilde{C} )$ is a homotopy equivalence of Kan complexes.

It will therefore suffice to show that, for every left fibration $q: \widetilde{C} \rightarrow C$, the restriction map $f^{\ast }: \operatorname{Fun}_{/C}(B, \widetilde{C} ) \rightarrow \operatorname{Fun}_{/C}( A, \widetilde{C} )$ is a homotopy equivalence if and only if the restriction map $(g \circ f)^{\ast }: \operatorname{Fun}_{/C}( C, \widetilde{C})\rightarrow \operatorname{Fun}_{/C}( A, \widetilde{C})$ is a homotopy equivalence. This is clear, since our assumption that $g$ is a categorical equivalence guarantees that the restriction map $g^{\ast }: \operatorname{Fun}_{/C}(C, \widetilde{C}) \rightarrow \operatorname{Fun}_{/C}(B, \widetilde{C} )$ is a homotopy equivalence (Corollary 7.2.1.12). $\square$

Corollary 7.2.1.21. Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{g} & B \ar [d]^{g'} \\ A' \ar [r]^-{f'} & B', }$

where $g$ and $g'$ are categorical equivalences. Then $f$ is left cofinal if and only if $f'$ is left cofinal.

Proof. By virtue of Proposition 7.2.1.20, the morphism $f$ is left cofinal if and only if the composite morphism $g' \circ f$ is left cofinal. Similarly, Proposition 7.2.1.6 guarantees that $f'$ is left cofinal if and only if $f' \circ g$ is left cofinal. We conclude by observing that $g' \circ f = f' \circ g$. $\square$

Corollary 7.2.1.22. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and suppose we are given a pair of diagrams $f_0, f_1: K \rightarrow \operatorname{\mathcal{C}}$ indexed by a simplicial set $K$. Suppose that $f_0$ and $f_1$ are isomorphic as objects of the $\infty$-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Then $f$ is left cofinal if and only if $g$ is left cofinal.

Proof. Let $\operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ be the full subcategory spanned by the isomorphisms of $\operatorname{\mathcal{C}}$ (see Example 4.4.1.13). Let $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the morphisms given by evaluation at the vertices $0,1 \in \Delta ^1$, so that $\operatorname{ev}_0$ and $\operatorname{ev}_1$ are trivial Kan fibrations (Corollary 4.4.5.10). Fix an isomorphism of $f_0$ with $f_1$, which we identify with a diagram $F: K \rightarrow \operatorname{Isom}(\operatorname{\mathcal{C}})$ satisfying $\operatorname{ev}_0 \circ F = f_0$ and $\operatorname{ev}_1 \circ F = f_1$. Applying Corollary 7.2.1.21 to the diagram

$\xymatrix@R =50pt@C=50pt{ K \ar [r]^-{F} \ar [d]^{\operatorname{id}} & \operatorname{Isom}(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{ev}_0 } \\ K \ar [r]^-{f_0} & \operatorname{\mathcal{C}}, }$

we deduce that $f_0$ is left cofinal if and only if $F$ is left cofinal. By the same reasoning, this is equivalent to the condition that $f_1$ is left cofinal. $\square$