The Stacks project

Bhargav Bhatt, private communication.

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

  1. If $v \circ u$ is fully faithful, then $u$ is fully faithful.

  2. If $u \circ v$ is fully faithful, then $v$ is fully faithful.

Proof. Proof of (2). Assume $u \circ v$ is fully faithful. Say we have $X$, $Y$ in $\mathcal{D}$. Then the natural composite map

\[ \mathop{Mor}\nolimits (X,Y) \to \mathop{Mor}\nolimits (v(X),v(Y)) \to \mathop{Mor}\nolimits (u(v(X)), u(v(Y))) \]

is a bijection, so $v$ is at least faithful. To show full faithfulness, we must show that the second map above is injective. But the adjunction between $u$ and $v$ says that

\[ \mathop{Mor}\nolimits (v(X), v(Y)) \to \mathop{Mor}\nolimits (u(v(X)), u(v(Y))) \to \mathop{Mor}\nolimits (u(v(X)), Y) \]

is a bijection, where the first map is natural one and the second map comes from the counit $u(v(Y)) \to Y$ of the adjunction. So this says that $\mathop{Mor}\nolimits (v(X), v(Y)) \to \mathop{Mor}\nolimits (u(v(X)), u(v(Y)))$ is also injective, as wanted. The proof of (1) is dual to this. $\square$


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