Interpreting a diagram in Borceux-Quinteiro's paper on enriched sheaves

Question

I am somewhat new to working with enriched categories, and have a question about how to interpret Definition 1.2 in Borceux-Quinteiro's paper A theory of enriched sheaves.

The authors consider a locally finitely presentable, closed symmetric monoidal category $\mathcal{V}$. For a small $\mathcal{V}$-category $\mathcal{C}$, an enriched Grothendieck topology on $\mathcal{C}$ is defined to be, for each object $c$ of $\mathcal{C}$, a family $\mathcal{T}(c)$ of subobjects of the $\mathcal{V}$-functor $\mathcal{C}(-,c)$ satisfying three conditions, one of which is:

(T2) For any $R \in \mathcal{T}(c)$, and any $f : G \to \mathcal{C}(d,c)$, where $G$ is an object of a dense generating family for $\mathcal{V}$, the functor $f^{-1}R \in \mathcal{T}(d)$, where $f^{-1}R$ denotes the pullback $\require{AMScd}$ \begin{CD} f^{-1}R @>>> \{G,R\}\\ @VVV @VVV\\ \mathcal{C}(-,d) @>>> \{G, \mathcal{C}(-,c)\}. \end{CD}

(Here $\{G,R\}$ denotes the cotensor of $R$ by $G$, and the right and bottom edges of the square are morphisms induced by $R \rightarrowtail \mathcal{C}(-,c)$ and $f$, respectively. It's not explicitly stated, but we evidently also assume that $\mathcal{C}$ is cotensored over $\mathcal{V}$.)

My question is whether I should interpret $f^{-1}R$ as an ordinary limit in the underlying category of the $\mathcal{V}$-category $[\mathcal{C}^\text{op}, \mathcal{V}]$, or as a conical limit in $[\mathcal{C}^\text{op}, \mathcal{V}]$?

I know that the former implies the latter as long as $\mathcal{C}$ is cotensored over $\mathcal{V}$, but it's not clear to me what the authors intended here, so I'm hoping someone with more experience can help me understand.

Answer 1

Ordinary limits in $\mathcal K_0$ and conical limits in $\mathcal K$ exist and coincide whenever $\mathcal K$ has copowers (=tensors); see Kelly's book, page 50.

In your case you want $[\mathcal{C}^\text{op}, \mathcal{V}]$ to have copowers (not $\mathcal C^\text{op}$), and that is always that case, independently of $\mathcal C$, since $\mathcal V$ is cocomplete and colimits in presheaf $\mathcal V$-categories are computed pointwise.

So taking the ordinary limit in $[\mathcal{C}^\text{op}, \mathcal{V}]_0$ or the conical one in $[\mathcal{C}^\text{op}, \mathcal{V}]$ gives you the same result.