Lurie define in HTT(Def. 1.1.1.6) a topological category as a enriched category over compactly generated (and weakly Hausdorff) topological spaces, but usually we define a topological category as lifting of sources (other definitions).

My question: it's equivalent the definition of topological category of Lurie with the usually?

Note: Lurie mention: "I take that definition for convenience" (Pg. 7), but he does not say anything about the connection with usual definition.