In the context of homotopy coherent category theory one wishes to nicely pair enriched category theory with homotopy theory for the case that the category being enriched over is a model category or at least a homotopical category.
A category enriched over such may be an enriched homotopical category.
The most obvious example is:
In this case, the cofibrant and fibrant replacements serve as a closed monoidal deformation retract. Closed monoidal homotopical categories that are not monoidal model categories seem surprisingly hard to come by, given how much stronger the axioms of a monoidal model category appear. One other example that can probably be extracted from the theory of homotopy tensor products in Shulman (below) is:
The homotopy category of a closed (symmetric) monoidal homotopical category is itself closed (symmetric) monoidal.
The definition appears as definition 15.3, p. 44 in Shulman: Homotopy limits and colimits in enriched homotopy theory, the proposition as proposition 15.4, p. 45.