The idea is that the homotopy category$Ho_C$ of a category $C$ which is enriched over a suitable monoidal and homotopical category $V$ is itself a $Ho_V$-enriched category.

Recall that for $V$ as above, $Ho_V$ is closed monoidal.

Proposition

With $C$ a $V$-homotopical category, $Ho_{C_0}$ is the underlying category of a $Ho_V$-enriched category.

Write $Ho_C$ for this $Ho_V$-enriched category. This is the enriched analogue of the homotopy category of $C$.

So schematically we have (with all of the above qualifiers suppressed):

$(C \in V-Cat) \Rightarrow (Ho_C \in Ho_V-Cat)$

Construction of the enriched homotopy category

For $C$ an enriched homotopical $V$-category as above, the $Ho_V$-category $Ho_C$ is constructed from the homotopy category $Ho_{C_0}$ of the ordinary category underlying $C$ by constructing a $Ho_V$-module structure, essentially following section 4.3.2 of Hovey: Model categories.