The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 4.24.3. Let $u$ be a left adjoint to $v$ as in Definition 4.24.1. Then

  1. $u$ is fully faithful $\Leftrightarrow \text{id} \cong v \circ u$.

  2. $v$ is fully faithful $\Leftrightarrow u \circ v \cong \text{id}$.

Proof. Assume $u$ is fully faithful. We have to show the adjunction map $X \to v(u(X))$ is an isomorphism. Let $X' \to v(u(X))$ be any morphism. By adjointness this corresponds to a morphism $u(X') \to u(X)$. By fully faithfulness of $u$ this corresponds to a morphism $X' \to X$. Thus we see that $X \to v(u(X))$ defines a bijection $\mathop{Mor}\nolimits (X', X) \to \mathop{Mor}\nolimits (X', v(u(X)))$. Hence it is an isomorphism. Conversely, if $\varphi : \text{id} \to v \circ u$ is a natural isomorphism, then $u$ is faithful for the trivial reason that the composition $\mathop{Mor}\nolimits (X,X') \to \mathop{Mor}\nolimits (u(X),u(X')) \to \mathop{Mor}\nolimits (v(u(X)),v(u(X')))$ is a bijection. Furthermore, $u$ is full: A preimage for a morphism $\gamma : u(X) \to u(X')$ is $u(\varphi _{X'}^{-1} \circ v(\varepsilon _{u(X')} \circ u(\varphi _{X'}) \circ \gamma ) \circ \eta _ X)$, where $\eta : \text{id}_ C \to v \circ u$ is the unit and $\varepsilon : u \circ v \to \text{id}_ D$ is the counit of the adjunction.

Part (2) is dual to part (1). $\square$


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