Baire category properties of function spaces with the Fell hypograph topology

For a Tychonoff space $X$ and a subspace $Ysubsetmathbb R$, we study Baire category properties of the space $C_{downarrow F}(X,Y)$ of continuous functions from $X$ to $Y$, endowed with the Fell hypograph topology. We characterize pairs $X,Y$ for which the function space $C_{downarrow F}(X,Y)$ is $infty$-meager, meager, Baire, Choquet, strong Choquet, (almost) complete-metrizable or (almost) Polish.