# Kerodon

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### 9.1.7 Orthogonality

In the $\infty$-categorical setting, it will often be useful to view the collection of solutions to a lifting problem as a space, rather than a set.

Construction 9.1.7.1. Suppose we are given a lifting problem

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

in an $\infty$-category $\operatorname{\mathcal{C}}$, given by a morphism $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. We let $\operatorname{Sol}( \sigma )$ denote the simplicial set $\{ \sigma \} \times _{ \operatorname{Fun}( Q, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}})$, where $Q \subset \Delta ^3$ is the simplicial subset described in Remark 9.1.5.2. We will refer to $\operatorname{Sol}( \sigma )$ as the space of solutions to the lifting problem $\sigma$.

Remark 9.1.7.2. In the situation of Construction 9.1.7.1, vertices of the simplicial set $\operatorname{Sol}(\sigma )$ can be identified with solutions to the lifting problem $\sigma$ (in the sense of Definition 9.1.5.1). In particular, the lifting problem $\sigma$ admits a solution if and only if $\operatorname{Sol}(\sigma )$ is nonempty.

Remark 9.1.7.3. In the situation of Construction 9.1.7.1, the restriction map $\operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( Q, \operatorname{\mathcal{C}})$ is an isofibration of $\infty$-categories (Corollary 4.4.5.3). Moreover, since $Q$ contains every vertex of $\Delta ^3$, it is also conservative (Theorem 4.4.4.4). It follows that the solution space $\operatorname{Sol}(\sigma )$ is a Kan complex (Corollary 4.4.3.21).

Definition 9.1.7.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, and let $f: A \rightarrow B$ and $g: X \rightarrow Y$ be morphisms of $\operatorname{\mathcal{C}}$. We will say that $f$ is left orthogonal to $g$ if, for every lifting problem $\sigma :$

$\xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur] & Y }$

in the $\infty$-category $\operatorname{\mathcal{C}}$, the solution space $\operatorname{Sol}(\sigma )$ is a contractible Kan complex. In this case, we will also say that $g$ is right orthogonal to $f$.

Example 9.1.7.5. Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then a morphism $f: A \rightarrow B$ is left orthogonal to a morphism $g: X \rightarrow Y$ in the $\infty$-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ if and only if every lifting problem

$\xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r]^-{u_0} & X \ar [d]^{g} \\ B \ar [r]^-{\overline{u}} \ar@ {-->}[ur] & Y }$

admits a unique solution: that is, there is a unique morphism $u: B \rightarrow X$ satisfying $u \circ f = u_0$ and $g \circ u = \overline{u}$.

Remark 9.1.7.6. Let $f$ and $g$ be morphisms in an $\infty$-category $\operatorname{\mathcal{C}}$. Then $f$ is left orthogonal to $g$ in $\operatorname{\mathcal{C}}$ if and only if it is right orthogonal to $g$ when regarded as a morphism of the opposite $\infty$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

Remark 9.1.7.7. Let $f$ and $g$ be morphisms in an $\infty$-category $\operatorname{\mathcal{C}}$. If $f$ is left orthogonal to $g$ (in the sense of Definition 9.1.7.4), then it is weakly left orthogonal to $g$ (in the sense of Definition 9.1.5.8). Beware that the converse is false (Exercise 9.1.7.8).

Exercise 9.1.7.8. Let $f: A \twoheadrightarrow B$ be a surjective function between sets, and let $g: X \hookrightarrow Y$ be an injective function between sets. Show that:

• The morphism $f$ is left orthogonal to $g$ (in the category of sets).

• The morphism $g$ is weakly left orthogonal to $f$.

• Unless either $f$ or $g$ is a bijection, the morphism $g$ is not left orthogonal to $f$.

Variant 9.1.7.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $S$ and $T$ be collections of morphisms of $\operatorname{\mathcal{C}}$. We say that $S$ is left orthogonal to $T$ if every morphism $f \in S$ is weakly left orthogonal to every morphism $g \in T$. In this case, we also say that $T$ is right orthogonal to $S$. In the special case where $S = \{ f\}$ is a singleton, we abbreviate this condition by saying that $f$ is left orthogonal to $T$, or $T$ is right orthogonal to $f$. In the special case $T = \{ g\}$ is a singleton, we abbreviate this condition by saying that $g$ is right orthogonal to $S$, or $S$ is left orthogonal to $g$.

To establish some elementary properties of Definition 9.1.7.4, it will be convenient to give an alternative description of the solution spaces $\operatorname{Sol}(\sigma )$.

Construction 9.1.7.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\sigma :$

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

be a lifting problem in $\operatorname{\mathcal{C}}$. Then $\sigma$ determines a pair of objects $\widetilde{B}$, $\widetilde{X}$ in the $\infty$-category $\operatorname{\mathcal{C}}_{A/ \, / Y }$ (see Remark 9.1.5.7). Let $K$ denote the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{A/ \, /Y} }( \widetilde{B}, \widetilde{X} )$. We then have a tautological map $K \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}_{A/ \, /Y}$, which we can identify with a diagram $\{ A\} \star (K \times \Delta ^1) \star \{ Y\} \rightarrow \operatorname{\mathcal{C}}$. Composing with the quotient map

$K \times \Delta ^3 \simeq K \times ( \{ A\} \star \Delta ^1 \star \{ Y\} ) \twoheadrightarrow \{ A\} \star (K \times \Delta ^1) \star \{ Y\} ,$

we obtain a morphism $K \rightarrow \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}})$, which factors through the simplicial subset $\operatorname{Sol}(\sigma ) \subseteq \operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})$ of Construction 9.1.7.1. We therefore obtain a comparison map $\theta : \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{A/ \, /Y} }( \widetilde{B}, \widetilde{X} ) \rightarrow \operatorname{Sol}( \sigma )$.

Proposition 9.1.7.11. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\sigma :$

9.14
$$\begin{gathered}\label{equation:solution-space-as-mapping-space} \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r] & X \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & Y } \end{gathered}$$

be a lifting problem in $\operatorname{\mathcal{C}}$. Then the comparison map

$\theta : \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{A/ \, /Y} }( \widetilde{B}, \widetilde{X} ) \rightarrow \operatorname{Sol}( \sigma )$

of Construction 9.1.7.10 is a homotopy equivalence of Kan complexes.

Proof. Corollary 4.6.6.9 supplies a categorical pullback diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{ A / \, / Y} ) \ar [r] \ar [d] & \operatorname{Fun}( \{ A \} \star \Delta ^1 \star \{ Y \} , \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}_{ A/ \, /Y} ) \ar [r] & \operatorname{Fun}( \{ A \} \star \operatorname{\partial \Delta }^1 \star \{ Y\} , \operatorname{\mathcal{C}}), }$

where the vertical maps are isofibrations (Corollary 4.4.5.3). Unwinding the definitions, we see that the comparison map $\theta$ is obtained by taking vertical fibers over the vertex corresponding to the pair $( \widetilde{B}, \widetilde{X} )$. Corollary 4.5.2.31 guarantees that $\theta$ is an equivalence of $\infty$-categories. Since the source and target of $\theta$ are Kan complexes (Remark 9.1.7.3), it is a homotopy equivalence (Example 4.5.1.13). $\square$

Warning 9.1.7.12. In the situation of Proposition 9.1.7.11, the comparison map $\theta$ need not be an isomorphism of simplicial sets. However, it is always bijective on $0$-simplices: vertices of both $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{A/ \, /Y} }( \widetilde{B}, \widetilde{X})$ and $\operatorname{Sol}( \sigma )$ can be identified with solutions to the lifting problem $\sigma$.

Corollary 9.1.7.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, so that $g$ can be identified with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. The following conditions are equivalent:

$(1)$

The morphism $g$ is right orthogonal to $f$ (in the sense of Definition 9.1.7.4).

$(2)$

For every morphism $\widetilde{f}: \widetilde{A} \rightarrow \widetilde{B}$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f} ) = f$, the object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ is $\widetilde{f}$-local (in the sense of Definition 9.1.1.1).

Corollary 9.1.7.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. If either $f$ or $g$ is an isomorphism, then $f$ is left orthogonal to $g$.

Proof. Without loss of generality, we may assume that $f$ is an isomorphism. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, so that we can identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Corollary 9.1.7.13, it will suffice to show that $\widetilde{X}$ is $\widetilde{f}$-local for every morphism $\widetilde{f}$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f} ) = f$. This is a special case of Example 9.1.1.2, since $\widetilde{f}$ is an isomorphism (Proposition 4.4.2.11). $\square$

Corollary 9.1.7.15. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing a morphism $g: X \rightarrow Y$ and a $2$-simplex

9.15
$$\begin{gathered}\label{equation:two-out-of-three-for-orthogonal} \xymatrix@R =50pt@C=50pt{ & B \ar [dr]^{ f'' } & \\ A \ar [ur]^{f'} \ar [rr]^{f} & & C. } \end{gathered}$$

Assume that $f'$ is left orthogonal to $g$. Then $f$ is left orthogonal to $g$ if and only if $f''$ is left orthogonal to $g$.

Proof. Assume that $f''$ is left orthogonal to $g$; we will show that $f$ is left orthogonal to $g$ (the proof of the converse is similar). Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, so that we can identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Corollary 9.1.7.13, it will suffice to show that the object $\widetilde{X}$ is $\widetilde{f}$-local, for every morphism $\widetilde{f}$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f} ) = f$. Since $\pi$ is a right fibration (Proposition 4.3.6.1), we can lift (9.15) to a diagram

$\xymatrix@R =50pt@C=50pt{ & \widetilde{B} \ar [dr]^{ \widetilde{f}'' } & \\ \widetilde{A} \ar [ur]^{ \widetilde{f}'} \ar [rr]^{ \widetilde{f}} & & \widetilde{C} }$

in the $\infty$-category $\operatorname{\mathcal{C}}_{/Y}$. Corollary 9.1.7.13 guarantees that the object $\widetilde{X}$ is both $\widetilde{f}'$-local and $\widetilde{f}''$-local, so the desired result follows from Remark 9.1.1.11. $\square$

Warning 9.1.7.16. In the situation of Corollary 9.1.7.15, if the morphisms $f$ and $f''$ are left orthogonal to $g$, then $f'$ need not be left orthogonal to $g$.

Corollary 9.1.7.17. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $g: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. If another morphism $f: A \rightarrow B$ is left orthogonal to $g$, then any retract of $f$ (in the $\infty$-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$) is also left orthogonal to $g$.

Proof. We proceed as in the proof of Proposition 9.1.5.14. Let $f': A' \rightarrow B'$ be a retract of $f$ (in the $\infty$-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$); we will show that $f'$ is left orthogonal to $g$. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let us identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Corollary 9.1.7.13, it will suffice to show that for any morphism $\widetilde{f}'$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f}' ) = f'$, the object $\widetilde{X}$ is $\widetilde{f}'$-local. It follows from Corollary 4.2.5.2 that $\pi$ induces a right fibration $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{/Y} ) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. Applying Remark 8.5.1.23, we deduce that $\widetilde{f}'$ is a retract of a morphism $\widetilde{f}$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $U( \widetilde{f} ) = f$. By virtue of Variant 9.1.1.7, it will suffice to show that the object $\widetilde{X}$ is $\widetilde{f}$-local, which follows from our assumption that $f$ is left orthogonal to $g$ (Corollary 9.1.7.13). $\square$

Corollary 9.1.7.18. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing a pushout diagram

9.16
$$\begin{gathered}\label{equation:pushout-of-left-orthogonal} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{f} & A' \ar [d]^{f'} \\ B \ar [r] & B'. } \end{gathered}$$

If $f$ is left orthogonal to a morphism $g: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, then $f'$ is also left orthogonal to $g$.

Proof. We proceed as in the proof of Proposition 9.1.5.15. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let us identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Corollary 9.1.7.13, it will suffice to show that for any morphism $\widetilde{f}'$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f}' ) = f'$, the object $\widetilde{X}$ is weakly $\widetilde{f}'$-local. Since $\pi$ is a right fibration, we can lift (9.16) to a diagram

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

in the $\infty$-category $\operatorname{\mathcal{C}}_{/Y}$, which is also a pushout square (Proposition 7.1.3.19). By virtue of Remark 9.1.1.10, it will suffice to show that $\widetilde{X}$ is $\widetilde{f}$-local, which follows from our assumption that $f$ is left orthogonal to $g$ (Corollary 9.1.7.13). $\square$

Corollary 9.1.7.19. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$ which is the colimit of a diagram

$Q_0: K \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \quad \quad v \mapsto f_{v}$

which is preserved by the evaluation functors $\operatorname{ev}_{0}, \operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$. Let $g: X \rightarrow Y$ be a morphism which is right orthogonal to each of the morphisms $f_{v}$. Then $g$ is right orthogonal to $f$.

Proof. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let us identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Corollary 9.1.7.13, it will suffice to show that if $\widetilde{f}$ is a morphism in $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f} ) = f$, then $\widetilde{X}$ is $\widetilde{f}$-local. Choose a colimit diagram $Q: K^{\triangleright } \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ which satisfies $Q|_{K} = Q_0$ and carries the cone point of $K^{\triangleright }$ to $f$. Since the inclusion of the cone point into $K^{\triangleright }$ is right anodyne (Example 4.3.7.11) and the projection map $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{/Y} ) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ is a right fibration (Corollary 4.2.5.2) we can lift $Q$ to a diagram $\widetilde{Q}: K^{\triangleright } \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ carrying the cone point to $\widetilde{f}$. Using Corollary 7.1.3.20 and Proposition 7.1.6.1, we see that $\widetilde{Q}$ is a colimit diagram which is preserved by the evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{/Y} ) \rightarrow \operatorname{\mathcal{C}}_{/Y}$. For each vertex $v \in K$, let $\widetilde{f}_{v} = \widetilde{Q}(v)$ is a morphism of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f}_{v} ) = f_{v}$. Our assumption that $f_{v}$ is left orthogonal to $g$ guarantees that $\widetilde{X}$ is a $\widetilde{f}_{v}$-local object of the $\infty$-category $\operatorname{\mathcal{C}}_{/Y}$ (Corollary 9.1.7.13). Applying Remark 9.1.1.9, we deduce that $\widetilde{X}$ is also $\widetilde{f}$-local. $\square$

Corollary 9.1.7.20. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $g: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $S$ be the collection of morphisms of $\operatorname{\mathcal{C}}$ which are left orthogonal to $g$. Then $S$ is weakly saturated.

Proof. Combining Corollaries 9.1.7.14, 9.1.7.15, and 9.1.7.19 with Proposition 9.1.2.10 (and Remark 9.1.2.11), we see that $S$ is closed under transfinite composition. Since $S$ is also closed under retracts (Corollary 9.1.7.17) and pushouts (Corollary 9.1.7.18), it is weakly saturated. $\square$

Corollary 9.1.7.21. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $g: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which is weakly right orthogonal to $S$. Assume that every morphism $f: A \rightarrow B$ which belongs to $S$ admits a relative codiagonal $\gamma _{A/B}: B \coprod _{A} B \rightarrow B$ which also belongs to $S$ (see Variant 7.6.3.19). Then $g$ is right orthogonal to $S$.

Proof. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let us identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. Let $\widetilde{S}$ denote the collection of those morphisms $\widetilde{f}$ in $\operatorname{\mathcal{C}}_{/Y}$ which satisfy $\pi ( \widetilde{f} ) \in S$. Our assumption that $g$ is weakly right orthogonal to $S$ guarantees that $\widetilde{X}$ is weakly $\widetilde{S}$-local (Remark 9.1.5.12). It follows from Proposition 7.1.3.19 that every morphism of $\widetilde{S}$ admits a relative codiagonal which also belongs to $\widetilde{S}$, so that $\widetilde{X}$ is $\widetilde{S}$-local (Proposition 9.1.3.15). Invoking Corollary 9.1.7.13, we conclude that $g$ is right orthogonal to $S$. $\square$

Proposition 9.1.7.22. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty$-categories, let $\sigma :$

9.17
$$\begin{gathered}\label{equation:solution-space-fibration} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur] & Y } \end{gathered}$$

be a lifting problem in the $\infty$-category $\operatorname{\mathcal{E}}$, and let $\overline{\sigma } = U \circ \sigma$ denote the associated lifting problem in the $\infty$-category $\operatorname{\mathcal{C}}$. If the morphism $f$ is $U$-cocartesian or the morphism $\widetilde{g}$ is $U$-cartesian, then $U$ induces a homotopy equivalence of solution spaces $\operatorname{Sol}( \sigma ) \rightarrow \operatorname{Sol}( \overline{\sigma } )$.

Proof. Let us identify the diagram (9.17) with a pair of objects $\widetilde{B}, \widetilde{X} \in \operatorname{\mathcal{E}}_{ A/ \, /Y}$. Note that $U$ induces a functor $\widetilde{U}: \operatorname{\mathcal{E}}_{ A/ \, /Y } \rightarrow \operatorname{\mathcal{C}}_{ U(A) / \, / U(Y) }$, and we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{\mathcal{E}}_{ A/ \, /Y}}( \widetilde{B}, \widetilde{X} ) \ar [r] \ar [d] & \operatorname{Sol}( \sigma ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{ U(A) / \, / U(Y) } }( \widetilde{U}( \widetilde{B} ), \widetilde{U}( \widetilde{X} ) ) \ar [r] & \operatorname{Sol}( \overline{\sigma } ), }$

where the horizontal maps are the homotopy equivalences supplied by Proposition 9.1.7.11. It will therefore suffice to show that the left vertical map is a homotopy equivalence. Without loss of generality, we may assume that the morphism $f$ is $U$-cocartesian. In this case, we will complete the proof by showing that the object $\widetilde{B} \in \operatorname{\mathcal{E}}_{ A/ \, /Y}$ is $\widetilde{U}$-initial. Let $U_{ / Y}: \operatorname{\mathcal{E}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/U(Y)}$ be the inner fibration induced by $U$; by virtue of Example 7.1.5.9, it will suffice to show that the lower left of the diagram (9.17) is $U_{/Y}$-cocartesian when viewed as a morphism in $\operatorname{\mathcal{E}}_{/Y}$. This follows from our assumption that $f$ is $U$-cocartesian (Corollary 5.1.1.14). $\square$

Corollary 9.1.7.23. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty$-categories and let $f$ and $g$ be morphisms of $\operatorname{\mathcal{E}}$. Assume either that $f$ is $U$-cocartesian or that $g$ is $U$-cartesian. Then:

• If $U(f)$ is left orthogonal to $U(g)$ in the $\infty$-category $\operatorname{\mathcal{C}}$, then $f$ is left orthogonal to $g$ in the $\infty$-category $\operatorname{\mathcal{E}}$.

• If $U(f)$ is weakly left orthogonal to $U(g)$ in the $\infty$-category $\operatorname{\mathcal{C}}$, then $f$ is weakly left orthogonal to $g$ in the $\infty$-category $\operatorname{\mathcal{E}}$.