The Stacks project

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

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

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 $\mathop{\mathrm{Mor}}\nolimits (v(X), v(Y)) \to \mathop{\mathrm{Mor}}\nolimits (u(v(X)), u(v(Y)))$ is also injective, as wanted. The proof of (1) is dual to this. $\square$