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However, they coincide when $\mathcal C$ is locally $\kappa$-presentable as a $\mathcal V$-category: so all the $\kappa$-presentable objects are $\kappa$-compact in the enriched sense. The reason for this is the following. Let us write $\mathcal A$ for the essentially small full subcategory of $\kappa$-presentable objects. Clearly $\mathcal A$ has $\kappa$-small colimits, and $\mathcal C$ is the free completion $\kappa\text-\mathbf{Filt}(\mathcal A)$ of $\mathcal A$ under conical $\kappa$-filtered colimits. But in fact, $\mathcal C$ is also the free completion $\kappa\text-\mathbf{Flat}(\mathcal A)$ of $\mathcal A$ under $\kappa$-flat colimits. Given this, a functor out of $\mathcal C$ preserves conical $\kappa$-filtered colimits iff it is the left Kan extension of its own restriction to $\mathcal A$, iff it preserves $\kappa$-flat colimits: in particular, $\kappa$-presentability and $\kappa$-compactness in $\mathcal C$ will coincide.

That $\kappa\text-\mathbf{Flat}(\mathcal A) = \kappa\text-\mathbf{Filt}(\mathcal A)$ is proven in Theorem 6.11 of Kelly's "Structures defined by...". Here's a brief summary. Firstly, $\kappa\text-\mathbf{Flat}(\mathcal A)$ is fairly easily seen to be the category of $\kappa$-continuous $\mathcal V$-functors $\mathcal A^{op} \to \mathcal{V}$, and is reflective in $[\mathcal A^{op},\mathcal V]$. But since every $X \in [\mathcal A^{op}, \mathcal V]$ can be written as a conical $\kappa$-filtered colimit of a $\kappa$-small colimit of representables, it follows that every $X \in \kappa\text-\mathbf{Flat}(\mathcal A)$ can also be so written; and since the representables in $\kappa\text-\mathbf{Flat}(\mathcal A)$ are closed under $\kappa$-small colimits, every $X \in \kappa\text-\mathbf{Flat}(\mathcal A)$ can be written as a conical $\kappa$-filtered colimit of representables. In particular, $\kappa\text-\mathbf{Flat}(\mathcal A)$ is the closure of the representables therein under conical $\kappa$-filtered colimits, but since these commute with $\kappa$-small limits, they are computed as in the whole presheaf category $[\mathcal A^{op},\mathcal V]$; so $\kappa\text-\mathbf{Flat}(\mathcal A)$ is equally the closure of the representables in $[\mathcal A^{op},\mathcal V]$ under conical $\kappa$-filtered colimits, which is precisely $\kappa\text-\mathbf{Filt}(\mathcal A)$.

However, they coincide when $\mathcal C$ is locally $\kappa$-presentable as a $\mathcal V$-category: so all the $\kappa$-presentable objects are $\kappa$-compact in the enriched sense. This is actually in the BQR paper you cite (Lemma 6.5) and follows from the following fact. Let us write $\mathcal A$ for the essentially small full subcategory of $\kappa$-presentable objects. Clearly $\mathcal A$ has $\kappa$-small colimits, and $\mathcal C$ is the free completion $\kappa\text-\mathbf{Filt}(\mathcal A)$ of $\mathcal A$ under conical $\kappa$-filtered colimits. But in fact, $\mathcal C$ is also the free completion $\kappa\text-\mathbf{Flat}(\mathcal A)$ of $\mathcal A$ under $\kappa$-flat colimits. Given this, a functor out of $\mathcal C$ preserves conical $\kappa$-filtered colimits iff it is the left Kan extension of its own restriction to $\mathcal A$, iff it preserves $\kappa$-flat colimits: in particular, $\kappa$-presentability and $\kappa$-compactness in $\mathcal C$ will coincide.

That $\kappa\text-\mathbf{Flat}(\mathcal A) = \kappa\text-\mathbf{Filt}(\mathcal A)$ is proven in Theorem 6.11 of Kelly's "Structures defined by...", or equally by Prop 4.5 of BQR (as you mention in your question.)

EDIT

I agree with Simon that if $\mathcal{C}$ is a cocomplete $\mathcal{V}$-category whose underlying category is locally presentable, then one can always find some $\kappa$ such that $\mathcal{C}$ is locally presentable as a $\mathcal{V}$-category, meaning that $\mathcal{C}_0$ is locally $\kappa$-presentable and the $\kappa$-presentable objects are closed under tensors by $\kappa$-presentable objects of $\mathcal{V}$.

Here, by saying that $X \in \mathcal C$ is $\kappa$-presentable, I just mean that $\mathcal C(X,\text{-}) \colon \mathcal{C} \rightarrow \mathcal V$ preserves conical filtered colimits (this is Kelly's definition). As Rune says, one could also talk of $X \in \mathcal C$ being $\kappa$-compact, meaning that $\mathcal C(X,\text{-})$ preserves $\kappa$-flat colimits. Since there is in no reason to believe that every $\kappa$-flat weight is a $\kappa$-filtered conical colimit of representables, these two notions will in general be distinct.

However, they coincide when $\mathcal C$ is locally $\kappa$-presentable as a $\mathcal V$-category: so all the $\kappa$-presentable objects are $\kappa$-compact in the enriched sense. The reason for this is the following. Let us write $\mathcal A$ for the essentially small full subcategory of $\kappa$-presentable objects. Clearly $\mathcal A$ has $\kappa$-small colimits, and $\mathcal C$ is the free completion $\kappa\text-\mathbf{Filt}(\mathcal A)$ of $\mathcal A$ under conical $\kappa$-filtered colimits. But in fact, $\mathcal C$ is also the free completion $\kappa\text-\mathbf{Flat}(\mathcal A)$ of $\mathcal A$ under $\kappa$-flat colimits. Given this, a functor out of $\mathcal C$ preserves conical $\kappa$-filtered colimits iff it is the left Kan extension of its own restriction to $\mathcal A$, iff it preserves $\kappa$-flat colimits: in particular, $\kappa$-presentability and $\kappa$-compactness in $\mathcal C$ will coincide.

That $\kappa\text-\mathbf{Flat}(\mathcal A) = \kappa\text-\mathbf{Filt}(\mathcal A)$ is proven in Theorem 6.11 of Kelly's "Structures defined by...". Here's a brief summary. Firstly, $\kappa\text-\mathbf{Flat}(\mathcal A)$ is fairly easily seen to be the category of $\kappa$-continuous $\mathcal V$-functors $\mathcal A^{op} \to \mathcal{V}$, and is reflective in $[\mathcal A^{op},\mathcal V]$. But since every $X \in [\mathcal A^{op}, \mathcal V]$ can be written as a conical $\kappa$-filtered colimit of a $\kappa$-small colimit of representables, it follows that every $X \in \kappa\text-\mathbf{Flat}(\mathcal A)$ can also be so written; and since the representables in $\kappa\text-\mathbf{Flat}(\mathcal A)$ are closed under $\kappa$-small colimits, every $X \in \kappa\text-\mathbf{Flat}(\mathcal A)$ can be written as a conical $\kappa$-filtered colimit of representables. In particular, $\kappa\text-\mathbf{Flat}(\mathcal A)$ is the closure of the representables therein under conical $\kappa$-filtered colimits, but since these commute with $\kappa$-small limits, they are computed as in the whole presheaf category $[\mathcal A^{op},\mathcal V]$; so $\kappa\text-\mathbf{Flat}(\mathcal A)$ is equally the closure of the representables in $[\mathcal A^{op},\mathcal V]$ under conical $\kappa$-filtered colimits, which is precisely $\kappa\text-\mathbf{Filt}(\mathcal A)$.

For Q1: this is dealt with in a context more general than the classical one by Adamek, Borceux, Lack and Rosicky in their paper "A classification of accessible categories". They replace finite or $\kappa$-small limits with an arbitrary class of limits $\mathbb{D}$, and consider a condition which they call soundness, one of whose consequences is a decomposition of every $\mathbb{D}$-flat weight as a suitably "$\mathbb{D}$-filtered" colimit of representables.

This is all in the unenriched context, which is not what you want, but the point is that they make axiomatic assumptions which are more or less exactly what is needed to force the answer to your question 1 to be true. Make of that what you will, but it at least implies that it's not automatic, and will probably require a bespoke argument in each situation.

For Q2: No. I guess the classical reference is Kelly's "Structures defined by finite limits in the enriched context." If $\mathcal V$ is a symmetric monoidal closed category which is locally $\kappa$-presentable as a closed category (i.e., it is locally $\kappa$-presentable and the $\kappa$-presentable objects are closed under the monoidal structure), then there is a good notion of locally $\lambda$-presentable $\mathcal V$-category: they are precisely the cocomplete $\mathcal V$-categories, whose underlying ordinary categories are locally $\lambda$-presentable, and whose $\lambda$-presentable objects are closed under tensors (=copowers) with $\lambda$-presentable objects of $\mathcal V$. Without this last condition, there is a gap through which to thread a negative answer to your question.

For Q1: something related is dealt with in a context more general than the classical one by Adamek, Borceux, Lack and Rosicky in their paper "A classification of accessible categories". They replace finite or $\kappa$-small limits with an arbitrary class of limits $\mathbb{D}$, and consider a condition which they call soundness, one of whose consequences is a decomposition of every $\mathbb{D}$-flat weight as a suitably "$\mathbb{D}$-filtered" colimit of representables.

This is all in the unenriched context, which is not what you want, but the point is that they make axiomatic assumptions which are more or less exactly what is needed to force the answer to your question 1 to be true. Make of that what you will, but it at least suggests that it's not automatic, and will probably require a bespoke argument in each situation.

For Q2: No. I guess the classical reference is Kelly's "Structures defined by finite limits in the enriched context." If $\mathcal V$ is a symmetric monoidal closed category which is locally $\kappa$-presentable as a closed category (i.e., it is locally $\kappa$-presentable and the $\kappa$-presentable objects are closed under the monoidal structure), then there is a good notion of locally $\lambda$-presentable $\mathcal V$-category: they are precisely the cocomplete $\mathcal V$-categories, whose underlying ordinary categories are locally $\lambda$-presentable, and whose $\lambda$-presentable objects are closed under tensors (=copowers) with $\lambda$-presentable objects of $\mathcal V$. Without this last condition, there is a gap through which to thread a negative answer to your question.