Let $M$ be a complex manifold with a Kahler metric $g$, define the covariant derivative of a smooth complex valued $T^{(1,0)}M$ vector field $X = X^i\partial_i$ to be such that its i^{th} component is $\nabla_jX = \partial_jX^i +\Gamma_{jk}^{i}X^j$, where $\Gamma_{jk}^{i} = g^{\bar{l}i}\partial_j g_{k\bar{l}}$. The claim is that the ith component of $\nabla_j X$ is a tensor.
I am confused why this is true. I believe that I need to verify it is $C^{\infty}(M)$ linear in order to show that it is a tensor. But $\partial_jfX^{i} +\Gamma_{jk}^{i}fX^j = \partial_j f X_i +f \partial_j X^i +f \Gamma_{jk}^{i}X^j$ which does not seem to be $C^{\infty}(M)$ linear.
Maybe the statement is that it is linear with respect to holomorphic functions but that still does not seem correct.