Definition of coevaluation map of a stable category defined in Tammo tom Dieck's Algebraic Topology.

Question

The following picture is from [Tammo tom Dieck, Algebraic Topology, pp.176].

Question

I want to ask if the statement that $j$ induces an h-equivalence :

$$C(V|K)\overset{}{\longrightarrow}C(\mathbb{R}^n|K)$$

is correct.

Attempt

In the book, the author stated

So as a first step, I want to show that $j:V\to\mathbb{R}^n$ is an h-equivalence, but I don't think it is possible to do so, since $\mathbb{R}^n$ is contractible but not all open subset of $\mathbb{R}^n$ has only one component or path component.