Proposition 7.2.1.10. 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.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
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 5.6.7.3 (and Remark 5.6.7.4), 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 of $A$ with respect to $W$ guarantees that the left horizontal maps are equivalences of $\infty $-categories. Applying Corollary 4.5.2.32, 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). $\square$