  For a Tychonoff space $X$ and a subspace $Y\subset\mathbb R$, 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.
