In Hartshorne's Algebraic Geometry II.8.20.1 (page 182), he takes the dual of Euler sequence
$$0 \rightarrow \Omega_{X/k} \rightarrow \mathcal{O}_{X}(-1)^{n+1} \rightarrow \mathcal{O}_{X} \rightarrow 0,$$
where $X = \mathbb{P}_{k}^{n}$, and get
$$0 \rightarrow \mathcal{O}_{X} \rightarrow \mathcal{O}_{X}(1)^{n+1} \rightarrow \mathscr{T}_{X} \rightarrow 0,$$
where $\mathscr{T}_{X} = \mathcal{Hom}(\Omega_{X/k}, \mathcal{O}_{X})$ is the tangent sheaf of $X$. But, is the dualizing functor $\mathcal{Hom}( \cdot, \mathcal{O}_{X})$ exact? Is not it only a left exact contravariant functor? Why in this case we have exactness of the sequence?