The Soergel functor is not fully faithful on all category-O objects

Exactness of $V$ applied to $0\to L(-2)\to M(0)\to L(0)\to 0$ gives $0\to V(L(-2))\to V(M(0))\to V(L(0))\to 0$, that is $0\to \mathbb{k}\to V(M(0))\to 0$, so $V(M(0))\cong \mathbb{k}$, and $V(P(-2))\cong C$; also $P(-2)$ has indecomposable head $L(-2)$ so $V(L(0))=\mathrm{Hom}(P(-2),L(0))=0$ and $V(L(-2))=\mathbb{k}$.

Non-faithfulness: $L(0)$ is a nonzero object with one-dimensional endomorphism ring, and ${\mathrm{id}}_{L(0)}\mathrm{\ne }0$ is a nonzero element of ${\mathrm{Hom}}_{\mathcal{O}}(L(0),L(0))=\mathbb{k}$, while $V(L(0))=0$ forces its image in ${\mathrm{Hom}}_C(0,0)=0$ to be zero; so $V$ is not faithful.

Failure of fullness: ${\mathrm{Hom}}_{\mathcal{O}}(M(0),L(-2))=0$ while $V(M(0))\cong V(L(-2))\cong \mathbb{k}$ and ${\mathrm{Hom}}_C(\mathbb{k},\mathbb{k})=\mathbb{k}\mathrm{\ne }0$, so the induced map between these Hom spaces is the zero map on a nonzero target and is not surjective.

The two failures use a nonprojective target: $L(0)$ is nonprojective because the nonsplit surjection $M(0)\to L(0)$ would otherwise split, and $L(-2)$ is nonprojective because the nonsplit surjection $P(-2)\to L(-2)$ would otherwise split, with $P(-2)$ indecomposable; hence they do not contradict the Struktursatz Soergel's Struktursatz: the functor V is fully faithful on projectives, whose target hypothesis is exactly that $Q$ be projective (the source may be arbitrary), and they do not refute the separate assertion of faithfulness on the quotient by the $V$-killed Serre subcategory, since $L(0)$ is precisely an object killed by $V$.