A deformation retract for the enrichment of a category enriched over a closed monoidal homotopical category is a deformation retract of a homotopical category for the two-variable adjunction
(power, copower, internal hom) :
satisfying a bunch of extra conditions.
This is [definition 16.1, p. 46] in Shulman: Homotopy limits and colimits and enriched homotopy theory
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