In [3] we defined a notion of compactness in FCS, the category of fuzzy convergence spaces as defined by Lowen/Lowen/Wuyts [8]. In their paper the latter also introduced a fuzzy convergence structure c-lim for fuzzy function spaces thus proving that FCS is a topological quasitopos. In this paper we start the investigation of compactness criteria of the Arzelà-Ascoli type in these fuzzy function spaces. To this aim we first give a coarser fuzzy convergence structure p-lim of pointwise convergence, for which compactness is easily established (via the Tychonoff theorem) and then secondly introduce a notion of evenly continuous fuzzy subsets on which p-lim and c-lim coincide.