The category of enriched functors from finite based CW complexes to based topological spaces has a projective model structure. The fibrations are the objectwise Serre fibrations and the weak equivalences are the objectwise weak homotopy equivalences. The cofibrations are defined by the left lifting property. In particular every cofibration is an objectwise Hurewicz cofibration of based spaces, but probably not an objectwise Serre cofibration.
Is this model structure cellular in the sense of Definition 12.1.1 of Hirschhorn's Model Categories and Their Localisations? It is known to be cofibrantly generated.