The aim of this article is to look at fuzzy topology from the viewpoint of categorical topology. Starting with FTS (the category of fuzzy topological spaces) [\textit{R. Lowen}, J. Math. Analysis Appl. 56, 621-633 (1976; Zbl 0342.54003)] the authors determine which subcategories of FTS, for instance, TOP (the category of topological spaces), FNS (the category of fuzzy neighborhood spaces), MAX (maximal, with respect to level topologies, fuzzy topological spaces) [\textit{P. Wuyts}, Fuzzy Sets Syst. 12, 71-85 (1984; Zbl 0574.54004)] are bireflective and/or coreflective. It is shown that the subcategories TOP and FNS are bireflective and coreflective, while MAX is a coreflective subcategory of FTS but fails to be bireflective (see the original paper for details). The most significant aspect of the present paper is the construction of FCS (the category of all fuzzy convergence spaces) the supercategory of FTS based on the theory of convergence in fuzzy topological spaces [\textit{R. Lowen}, General Topol. Appl. 10, 147-160 (1979; Zbl 0409.54008)]. The authors successfully prove that FTS is a bireflective subcategory of FCS, and most importantly, FCS is, in contrast to FTS, a topological quasitopos which means it is at the same time cartesian closed and extensional [\textit{L. D. Nel}, Categor. Topol., Proc. Conf. Mannheim 1975, Lect. Notes Math. 540, 439-451 (1976; Zbl 0336.54006)]. Finally, they observed that PSTOP (the category of pseudotopological convergence spaces) is a simultaneously coreflective and bireflective subcategory of FCS.