My inquiry pertains to the reference [1]. In section 2.3 on page 7 of this paper, the author asserts that if $u\in \Gamma(U,\Omega_{(2)}^{p,q}(X,E))$ and $f\in C^\infty (X)$, then $fu\in \Gamma(U,\Omega_{(2)}^{p,q}(X,E))$, thus demonstrating the existence of a partition of unity in $\Omega_{(2)}^{p,q}(X,E)$. Subsequently, the author claims that $\Omega_{(2)}^{p,q}(X,E)$ is a fine sheaf over $X$. What is the basis for this assertion? I'm very confused.
[1] Huang, C., Liu, K., Wan, X., & Yang, X. (2016). Logarithmic vanishing theorems on compact K"{a} hler manifolds I. arXiv preprint arXiv:1611.07671.