For a closed monoidal homotopical category, a -enriched category with powers and copowers and with the structure of a homotopical category on its underlying category is a -homotopical category when equipped with a deformation retract for the enrichment.
Recall that for as above, is closed monoidal.
With a -homotopical category, is the underlying category of a -enriched category.
Write for this -enriched category. This is the enriched analogue of the homotopy category of .
So schematically we have (with all of the above qualifiers suppressed):
(C \in V-Cat) \Rightarrow (Ho_C \in Ho_V-Cat)
For an enriched homotopical -category as above, the -category is constructed from the homotopy category of the ordinary category underlying by constructing a -module structure, essentially following section 4.3.2 of Hovey: Model categories.
The definition appears as definition 16.1, p. 46 in Shulman: Homotopy limits and colimits in enriched homotopy theory, the proposition is proposition 16.2, p. 46. The construction of follows the proof of proposition 15.4, p. 45.