# Kerodon

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

### 9.1.9 Uniqueness of Factorizations

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $( S_{L}, S_{R} )$ be a weak factorization system on $\operatorname{\mathcal{C}}$ (Definition 9.1.6.1). Then every morphism $h: X \rightarrow Z$ admits a factorization

9.18
$$\begin{gathered}\label{equation:uniqueness-factorization} \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z, } \end{gathered}$$

where $f$ belongs to $S_{L}$ and $g$ belongs to $S_ R$. Our goal in this section is to show that the pair $(S_ L, S_ R)$ is a factorization system (in the sense of Definition 9.1.8.1) if and only if the diagram (9.18) is essentially unique (Theorem 9.1.9.2).

Notation 9.1.9.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$. We let $\operatorname{Fun}_{L}(\Delta ^2, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ spanned by those diagrams (9.18) where $f$ belongs to $S_{L}$, and $\operatorname{Fun}_{R}( \Delta ^2, \operatorname{\mathcal{C}})$ the full subcategory of $\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ spanned by those diagrams where $g$ belongs to $S_{R}$. We let $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ denote the intersection $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}}) \cap \operatorname{Fun}_{R}( \Delta ^2, \operatorname{\mathcal{C}})$.

Theorem 9.1.9.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which contain all identity morphisms and which are closed under isomorphism (in the $\infty$-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$). The following conditions are equivalent:

$(1)$

The pair $(S_{L}, S_{R} )$ is a factorization system on $\operatorname{\mathcal{C}}$.

$(2)$

Composition with the inclusion map $\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \hookrightarrow \Delta ^2$ induces an equivalence of $\infty$-categories

$D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma ).$

Remark 9.1.9.3. In the situation of Theorem 9.1.9.2, our hypothesis that the collections $S_{L}$ and $S_{R}$ are closed under isomorphism guarantees that the full subcategory $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ is replete, so that the functor $D$ is an isofibration of $\infty$-categories (see Corollary 4.4.5.3). Invoking Proposition 4.5.5.20, we see that $(2)$ is equivalent to the following a priori stronger condition:

$(2')$

The restriction functor

$D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma )$

is a trivial Kan fibration of $\infty$-categories.

Corollary 9.1.9.4 (Exponentiation of Factorization Systems). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category equipped with a factorization system $(S_ L, S_ R)$ and let $K$ be a simplicial set. Then the $\infty$-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ admits a factorization system $(S_ L^{K}, S_ R^{K})$, where $S_{L}^{K}$ denotes the collection of all morphisms $f$ in $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ such that $f(v) \in S_ L$ for each vertex $v$ of $K$, and $S_{R}^{K}$ is defined similarly.

Proof. Since $S_{L}$ and $S_{R}$ contain identity morphisms and are closed under isomorphism, the collections $S_{L}^{K}$ and $S_{R}^{K}$ have the same properties. By virtue of Theorem 9.1.9.2, it will suffice to show that the restriction map

$\theta : \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{Fun}(K, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{Fun}(K, \operatorname{\mathcal{C}}) ) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma )$

is an equivalence of $\infty$-categories. This follows from Remark 4.5.1.16, since $\theta$ is obtained by applying the functor $\operatorname{Fun}(K, \bullet )$ to the restriction map $\operatorname{Fun}_{LR}(\Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. $\square$

The proof of Theorem 9.1.9.2 will require some preliminaries. We begin by giving another description of the space of solutions to a lifting problem.

Notation 9.1.9.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. We let ${}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g}$ denote the the iterated fiber product

$\{ f\} \times _{ \operatorname{Fun}( \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) } \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \times _{\operatorname{Fun}_{ \operatorname{N}_{\bullet }( \{ 2 < 3 \} )}} \{ g\} ,$

whose objects can be identified with diagrams

$\xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [ur] \ar [r] & Y }$

in the $\infty$-category $\operatorname{\mathcal{C}}$.

Lemma 9.1.9.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. Then precomposition with the inclusion map $\operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \hookrightarrow \Delta ^3$ induces a trivial Kan fibration of simplicial sets ${}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, X )$. In particular, the simplicial set ${}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g}$ is a Kan complex.

Proof. By construction, we have a pullback diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \empty {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \ar [d] \ar [r] & \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B,X) \ar [r] & \operatorname{Fun}( \operatorname{Spine}[3], \operatorname{\mathcal{C}}), }$

where the right vertical map is a trivial Kan fibration (see Example 1.4.7.7). $\square$

Notation 9.1.9.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. We let $\widetilde{f} = s^{1}_{f}(f)$ and $\widetilde{g} = s^{1}_{0}(g)$ denote the degenerate $2$-simplices of $\operatorname{\mathcal{C}}$ depicted in the diagram

$A \xrightarrow {f} B \xrightarrow {\operatorname{id}} B \quad \quad X \xrightarrow {\operatorname{id}} X \xrightarrow {g} Y.$

Let $\beta : \Delta ^2 \times \Delta ^1 \rightarrow \Delta ^3$ denote the morphism of simplicial sets given on vertices by the formulae

$\beta (0,0) = 0 \quad \quad \beta (1,0) = 1 = \beta (2,0) \quad \quad \beta (0,1) = 2 = \beta (1,1) \quad \quad \beta (2,1) = 3.$

Then precomposition with $\beta$ determines a functor $\operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^2 \times \Delta ^1, \operatorname{\mathcal{C}})$, which restricts to a map of Kan complexes $T: {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} )$. Concretely, $T$ carries a $3$-simplex

$\xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r]^-{u} & X \ar [d]^{g} \\ B \ar [ur]^{v} \ar [r]^-{w} & Y }$

to the morphism in $\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ depicted in the diagram

$\xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{u} & B \ar [d]^{v} \ar [r]^-{\operatorname{id}_{B}} & B \ar [d]^{w} \\ X \ar [r]^-{\operatorname{id}_{X}} & X \ar [r]^-{g} & Y. }$

Lemma 9.1.9.8. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. Then the comparison map

$T: {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} )$

of Notation 9.1.9.7 is a homotopy equivalence.

Proof. By construction, the diagram $\widetilde{f}: \Delta ^2 \rightarrow \operatorname{\mathcal{C}}$ is left Kan extended from the simplicial subset $\Delta ^1 \subset \Delta ^2$. Applying Corollary 7.3.6.9, we deduce that the restriction functor $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ determines a trivial Kan fibration $R: \operatorname{Hom}_{\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f, \operatorname{id}_{X} )$. Similarly, the $1$-simplex $\operatorname{id}_{X}$ can be viewed as a diagram $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ which is right Kan extended from the vertex $\{ 1\} \subset \Delta ^2$, so the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ induces a trivial Kan fibration $Q: \operatorname{Hom}_{\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f, \operatorname{id}_{X} ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B, X )$. We are therefore reduced to showing that the composite map

${}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \xrightarrow {T} \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} ) \xrightarrow {R} \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f, \operatorname{id}_{X} ) \xrightarrow {Q} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, X )$

is a homotopy equivalence, which follows from Lemma 9.1.9.6. $\square$

In the situation of Lemma 9.1.9.8, restriction to the “long edge” of $\Delta ^2$ determines an inner fibration of $\infty$-categories $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (Corollary 4.1.4.2), and therefore induces a Kan fibration of mapping spaces

$\operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }( f, g ).$

Lemma 9.1.9.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing a lifting problem $\sigma :$

9.19
$$\begin{gathered}\label{equation:other-comparison-map-for-Sol} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur]^{s} & Y, } \end{gathered}$$

which we identify with a morphism from $f$ to $g$ in the $\infty$-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. Then the comparison map $T: {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} )$ of Notation 9.1.9.7 restricts to a homotopy equivalence of Kan complexes

$T_0: \operatorname{Sol}( \sigma ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} ) \times _{ \operatorname{Hom}_{\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f,g) } \{ \sigma \} .$

Proof. Let $\alpha : \Delta ^1 \times \Delta ^1 \hookrightarrow \Delta ^3$ be the morphism of simplicial sets given on vertices by the formula $\alpha (i,j) = 2i+j$ (see Definition 9.1.5.1). Then precomposition with $\alpha$ induces an isofibration of $\infty$-categories $U: \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{C}})$, which restricts to an isofibration $U_0: {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }( f, g )$. Since the source and target of $U_0$ are Kan complexes, it is a Kan fibration (Corollary 4.4.3.10). The desired result now follows by applying Corollary 3.3.7.3 to the diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ \empty {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \ar [dr]^{U_0} \ar [rr]^{T} & & \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} ) \ar [dl] \\ & \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }( f, g), }$

since $T$ is a homotopy equivalence (Lemma 9.1.9.8). $\square$

Corollary 9.1.9.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing $2$-simplices $\sigma$ and $\tau$. Suppose that the initial edge $f = d^{2}_{2}(\sigma )$ is left orthogonal to the final edge $g = d^{2}_{0}(\tau )$. Then the restriction map $\theta : \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }(\sigma , \tau ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }(f,g)$ is a trivial Kan fibration.

Proof. By virtue of Proposition 3.3.7.4, it will suffice to show that every fiber of $\theta$ is a contractible Kan complex. Let $D: \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ denote the functor given by precomposition with the inclusion map $\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \hookrightarrow \Delta ^2$, and let $E: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be given by evaluation at the final vertex $1 \in \Delta ^1$. Let $\gamma : \Delta ^2 \times \Delta ^1 \rightarrow \Delta ^2$ be the morphism of simplicial sets given on vertices by the formula $\gamma (i,j) = \begin{cases} 1 & \text{ if } (i,j) = (0,2) \\ i & \text{ otherwise. } \end{cases}$. Then the composition

$\Delta ^2 \times \Delta ^1 \xrightarrow {\gamma } \Delta ^2 \xrightarrow {\sigma } \operatorname{\mathcal{C}}$

can be regarded as a morphism $e: \widetilde{f} \rightarrow \sigma$ in the $\infty$-category $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$. It follows from Corollary 5.3.7.5 that the morphism $e$ is $(E \circ D)$-cocartesian and that $D(e)$ is $E$-cocartesian. Consequently, the morphism $e$ is $D$-cocartesian (Corollary 5.1.2.6). Using Proposition 5.1.2.1, we deduce that every fiber of $\theta$ is homotopy equivalent to a fiber of the restriction map $\theta ': \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }(\widetilde{f}, \tau ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }(f,g)$. It will therefore suffice to prove Corollary 9.1.9.10 in the special case where $\sigma = \widetilde{f}$. By a similar argument, we may also assume that $\tau$ is the degenerate $2$-simplex $\widetilde{g} = s^{1}_{0}(g)$. In this case, Lemma 9.1.9.9 guarantees that every fiber of $\theta$ is homotopy equivalent to the space of solutions to some lifting problem

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

which is contractible by virtue of our assumption that $f$ is left orthogonal to $g$. $\square$

Corollary 9.1.9.11. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which contain all identity morphisms. The following conditions are equivalent:

$(1)$

The restriction functor

$D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma )$

is fully faithful.

$(2)$

The collection $S_{L}$ is left orthogonal to $S_{R}$.

Proof. Assume that condition $(1)$ is satisfied; we will prove $(2)$ (the reverse implication follows immediately from Corollary 9.1.9.10). Suppose we are given a lifting problem $\tau :$

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

in the $\infty$-category $\operatorname{\mathcal{C}}$, where $f$ belongs to $S_{L}$ and $g$ belongs to $S_{R}$. We wish to show that the solution space $\operatorname{Sol}(\tau )$ is contractible. Let $\widetilde{f} = s^{1}_{f}(f)$ and $\widetilde{g} = s^{1}_{0}(g)$ denote the degenerate $2$-simplices of $\operatorname{\mathcal{C}}$ defined in Notation 9.1.9.5. Since $S_{L}$ and $S_{R}$ contain all identity morphisms, we can view $\widetilde{f}$ and $\widetilde{g}$ as objects of the $\infty$-category $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$. Assumption $(1)$ guarantees that the Kan fibration $\operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }( f,g )$ is a homotopy equivalence. Lemma 9.1.9.9 supplies a homotopy equivalence of $\operatorname{Sol}( \tau )$ with the fiber $\operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} )_{\tau }$, which is contractible by virtue of Proposition 3.3.7.4. $\square$

Proof of Theorem 9.1.9.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$. Assume first that the pair $(S_{L}, S_{R})$ is a factorization system on $\operatorname{\mathcal{C}}$; we wish to show that the restriction functor

$D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma )$

is an equivalence of $\infty$-categories. Using the assumption that $(S_{L}, S_{R} )$ is a weak factorization system, we deduce that the functor $D$ is surjective on objects; in particular, it is essentially surjective. By virtue of Theorem 4.6.2.19, we are reduced to proving that $D$ is fully faithful, which is a special case of Corollary 9.1.9.11.

We now prove the converse. Assume that $D$ is an equivalence of $\infty$-categories; we wish to show that the pair $(S_ L, S_ R)$ is a factorization system on $\operatorname{\mathcal{C}}$. Using Remark 9.1.9.3, we see that $D$ is a trivial Kan fibration. It therefore admits a section $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$, which carries each morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$ to a $2$-simplex $\sigma _{f}:$

$\xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{ f_{R} } & \\ X \ar [ur]^{ f_{L} } \ar [rr]^{f} & & Z, }$

where $f_{L} \in S_{L}$ and $f_{R} \in S_{R}$. Using Corollary 9.1.9.11, we deduce that $S_{L}$ is left orthogonal to $S_{R}$. We will complete the proof by showing that $S_{L}$ is closed under retracts (a similar argument shows that $S_{R}$ is closed under retracts). For this, it will suffice to show that a morphism $f$ of $\operatorname{\mathcal{C}}$ belongs to $S_{L}$ if and only $f_{R}$ is an isomorphism in $\operatorname{\mathcal{C}}$. One direction is clear: if $f_{R}$ is an isomorphism, then $f$ is isomorphic to $f_{L}$ in the $\infty$-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, and therefore belongs to $S_{L}$ by virtue of our assumption that $S_ L$ is closed under isomorphism. For the converse, assume that $f$ belongs to $S_{L}$. Then the degenerate $2$-simplex $\widetilde{f} = s^{1}_{1}(f)$ can be regarded as an object of $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ satisfying $D( \widetilde{f} ) = f = D( \sigma _{f} )$. Since $D$ is an equivalence of $\infty$-categories, the $2$-simplex $\sigma _{f}$ is isomorphic to $\widetilde{f}$ as an object of the $\infty$-category $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$. It follows that $f_{R} = d^{2}_{0}( \sigma _{f} )$ is isomorphic to $\operatorname{id}_{Z} = d^{2}_{0}( \widetilde{f} )$ as an object of the $\infty$-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, so that $f_{R}$ is an isomorphism (Example 4.4.1.13). $\square$