Let $C$ be a $V$-model category, and $\mathcal{K}$ a set of objects of $C$.
Let me denote (derived) simplicial homotopy function complexes by $\text{Dmap}$
and derived $V$-function complexes by $\text{DMap}\_V$.


*Edit*: By the latter, I just mean that $\text{DMap}\_V$ is obtained from the
$V$-enrichment $\text{Map}\_V$ on $M$ by using cofibrant/fibrant
replacements with respect to the given model structure on $C$, i.e.,
$\text{DMap}\_V (X,Y) = \text{Map}\_V (X^{\text{cof}},Y^{\text{fib}})$. 


Under some technical assumptions (combinatoriality and right properness of $C$
are sufficient), one can form the $V$-enriched colocalization of $C$ at $\mathcal{K}$,
denoted by $R^V\_{\mathcal{K}} C$ whose weak equivalences are called
$\mathcal{K}$-colocal equivalences.


Let us call the cofibrant objects of $R^V\_{\mathcal{K}} C$
*colocal objects*. There are two possible characterizations of
colocal objects, namely


1. $A$ is colocal if and only if
	1. $A$ is cofibrant with respect to the original model structure, and
	2. For every $\mathcal{K}$-colocal equivalence $h: X \to Y$, the induced map
	$\text{Dmap}(A,h): \text{Dmap}(A,X) \to \text{Dmap}(A,Y)$ is a weak
	equivalence of simplicial sets


**or**


2. $A$ is colocal if and only if
	1. $A$ is cofibrant with respect to the original model structure, and
	2. For every $\mathcal{K}$-colocal equivalence $h: X \to Y$, the induced
	map $\text{DMap}\_V (A,h): \text{DMap}\_V (A,X) \to \text{DMap}\_V (A,Y)$ is a
	weak equivalence in $V$.


*Edit: Made 1. and 2. more precise.*


Question: Why is 1. equivalent to 2.?
-------------------------------------


*In more detail:*


One construction of $R^V\_{\mathcal{K}} C$ is given as the
classical Bousfield colocalization $R\_{K \otimes \mathcal{G}\_V} C$ of
$C$ at the set $K \otimes \mathcal{G}\_V$ where $\otimes: C \times V \to C$
is part of the $V$-enrichment on $C$ and $\mathcal{G}\_V$ is a set of cofibrant
homotopy generators of $V$.


This is Definition 2.10 of
<https://arxiv.org/abs/1411.0500>, and the weak equivalences in $R^V\_{\mathcal{K}}
C$ are those maps $h: X \to Y$ such that the induced maps
$\text{Dmap}(K \otimes G, X) \to \text{Dmap} (K \otimes G,Y)$ are weak
equivalences of simplicial sets for $K \in \mathcal{K}, G \in \mathcal{G}\_V$
by definition. However, Theorem 2.12 shows that the $\mathcal{K}$-colocal equivalences 
are precisely those maps $h: X \to Y$ that induce weak
equivalences $\text{DMap}\_V (K,X) \to \text{DMap}\_V (K,Y)$ in $V$ for all $K \in
\mathcal{K}$.


Thus, I would like to have a result similar to Theorem 2.12 for the two
characterizations of cofibrant objects 1. and 2. from before.
Characterization 1. is immediate from the construction and results in
Hirschhorn's book.


*Edit: Removed a lot of text concerning another article. I have now
understood the notation in this article correctly, and my former elaborations
do not make sense anymore.*



Colocal Objects in Enriched Bousfield Colocalizations
It is not true that 1. is equivalent to 2.


We have that 2. implies 1. by one of Dmitri's comments to the question.


As Dmitri pointed out in the same comment, a counterexample to 1.=>2. can be constructed by a map of stacks that is not a weak equivalence but a weak equivalence at the point.


More precisely, we consider the enriched colocalization of the category of stacks at the object $\ast$. This procedure does not change the model category structure. The unique map from $\Omega^0\_{\text{cl}}$, the levelwise discrete stack of closed $0$-forms, to the stack $\ast$ represented by a one-point manifold, which is the terminal object in stacks, is an equivalence at the point because there is a unique $0$-form on the point, which is closed. But the map $\Omega^0\_{\text{cl}} \to \ast$ cannot be a weak equivalence of stacks because the underlying homotopy type of $\Omega^0\_{\text{cl}}$ is a $K(\mathbb{R}\_\delta,0)$ which is not trivial.


