Proposition 6.2.2.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion map. Let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a functor of $\infty $-categories and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \iota \circ L$ be a natural transformation. The following conditions are equivalent:
- $(1)$
The natural transformation $\eta $ is the unit of an adjunction: that is, it exhibits $L$ as a left adjoint to the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$.
- $(2)$
The natural transformation $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor: that is, for every object $X \in \operatorname{\mathcal{C}}$, the morphism $\eta _{X}: X \rightarrow L(X)$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$.
- $(3)$
For every object $X \in \operatorname{\mathcal{C}}$, the morphism $L( \eta _ X ): L(X) \rightarrow L(L(X))$ is an isomorphism in $\operatorname{\mathcal{C}}'$. Moreover, if $X$ belongs to $\operatorname{\mathcal{C}}'$, then $\eta _{X}: X \rightarrow L(X)$ is an isomorphism.
Moreover, if these conditions are satisfied, then any natural transformation $\epsilon : L \circ \iota \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}'}$ which is compatible with $\eta $ up to homotopy (in the sense of Definition 6.2.1.1) is an isomorphism in the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}', \operatorname{\mathcal{C}}')$.
Proof.
We first show that $(1)$ implies $(2)$. Let $X$ be an object of $\operatorname{\mathcal{C}}$, so that $\eta $ determines a morphism $\eta _{X}: X \rightarrow L(X)$. For every object $Y \in \operatorname{\mathcal{C}}'$, Proposition 6.2.1.17 guarantees that composition with the homotopy class $[\eta _ X]$ induces an isomorphism
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( L(X), Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( L(X), Y) \xrightarrow { \circ [ \eta _ X ] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Y) \]
in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. It follows that $\eta _{X}$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. Allowing $X$ to vary, we conclude that $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor.
We now show that $(2)$ implies $(3)$. Assume that, for every object $X \in \operatorname{\mathcal{C}}$, the morphism $\eta _ X: X \rightarrow L(X)$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. Note that we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{\eta _ X} \ar [d]^{\eta _ X} & L(X) \ar [d]^{ \eta _{L(X) }} \\ L(X) \ar [r]^-{ L( \eta _ X ) } & L(L(X)) } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, obtained by applying the natural transformation $\eta $ to the morphism $\eta _{X}: X \rightarrow L(X)$. For each object $Y \in \operatorname{\mathcal{C}}$, we obtain a commutative diagram of sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Y) & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( L(X), Y) \ar [l]_{ \circ [\eta _ X] } \\ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(L(X), Y) \ar [u]_{ \circ [ \eta _ X ] } & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( L(L(X)), Y). \ar [u]_{ \circ [ \eta _{L(X)} ] } \ar [l]_{ \circ [ L(\eta _ X) ] } } \]
If $Y$ belongs to the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$, then the vertical maps and the upper horizontal map in this diagram are bijective. It follows that the lower horizontal map is bijective as well. Allowing $Y$ to vary, we deduce that the homotopy class $[ L(\eta _ X) ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}'$, so that $L( \eta _ X )$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}'$. In the special case where $X$ belongs to $\operatorname{\mathcal{C}}'$, Example 6.2.2.4 guarantees that $\eta _ X$ is already an isomorphism before applying the functor $L$.
We now show that $(3)$ implies $(1)$. Note that $\eta $ determines natural transformations
\[ \eta ': L \rightarrow L \circ \iota \circ L \quad \quad (X \in \operatorname{\mathcal{C}}) \mapsto ( L(\eta _ X) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(L(X), L(L(X)) ) ) \]
\[ \eta '': \iota \rightarrow \iota \circ L \circ \iota \quad \quad (Y \in \operatorname{\mathcal{C}}') \mapsto (\eta _ Y \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, L(Y) ) ). \]
If condition $(3)$ is satisfied, then Theorem 4.4.4.4 guarantees that $\eta '$ and $\eta ''$ are isomorphisms in the $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}' )$ and $\operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{C}})$, respectively. Invoking the criterion of Proposition 6.1.4.6, we conclude that $\eta $ is the unit of an adjunction.
$\square$