Recently, R. Garner [Theory Appl. Categ. 29, 406–421 (2014; Zbl 1305.18005)] stated that a faithful functor is topological in the sense of categorical topology [J. Adámek et al., Repr. Theory Appl. Categ. 2006, No. 17, 1–507 (2006; Zbl 1113.18001)] if, and only if, it is total (same as totally cocomplete) in the sense of enriched category theory [R. Street and R. Walters, J. Algebra 50, 350–379 (1978; Zbl 0401.18004)], arriving thus at a common ground for two seemingly unrelated lines of category theory. Motivated by this achievement, the authors of the present paper make another step in the same direction, putting some results of categorical topology into the context of categories enriched in a quantaloid. In particular, they show how concrete categories over a category B can be naturally described as categories enriched in the free quantaloid over B. Interpreting structured sinks as presheaves gives then the result of R. Garner (Theorem 3.2 on page 217). Moreover, the authors provide an enriched category theory analogue of the result of O. Wyler [General Topology Appl. 1, 17–28 (1971; Zbl 0215.51502)] stating that a concrete category is topological if, and only if, it is a fibration and an opfibration, and has complete lattices as fibres (Corollary 6.3 on page 225). They then present an enriched analogue of the result of R.-E. Hoffmann [“Die kategorielle Auffassung der Initial- und Finaltopologie”, Ph.D. thesis, University of Bochum, (1972)] on self-duality for topological functors, and also show that a category is total precisely when it appears as the category of the fixed objects under the so-called Isbell adjunction induced by a distributor L. Shen and D. Zhang [Theory Appl. Categ. 28, 577–615 (2013; Zbl 1273.18022)] (cf. Theorem 8.5 on page 231, and Corollary 9.3 on page 233).
The paper is well written, carefully provides most of its required preliminaries, and will be of interest to the researchers, dealing with categorical topology.