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Premises:

Let $\textbf{V}$ a closed monoidal closed category, let ${}^{c}\textbf{V}$ its structure of $\textbf{V}$-category, given a $\textbf{V}$-category $\mathscr{A}$, we call it small if the class of its objects is a set, and let $|\mathscr{A}|$ its underling category (where $|\mathscr{A}|(A, B)$ is identified with $\textbf{V}(I, [A, B])$ ). Let $U_I: \textbf{V} \to Set: V \mapsto [I, A]$. We suppose that $U_I$ has a left-adjoint $L: Set \to \textbf{V}: S \mapsto S\cdot I$ where $S \cdot I:= \coprod_{s \in S} I_s $, $I_s=s\ s \in S$. Then given a category $\mathcal{C}$ there is the $\textbf{V}$-category $L_* \mathcal{C}$ and for each functor $F: \mathcal{C} \to |\mathscr{A}|$ there is unique $\widetilde{F}: L_* \mathcal{C}\to \mathscr{A} $ such that $|L_* F| \circ H_\mathcal{C}= F$ where $H_\mathcal{C}: C \to |L_*\mathcal{C}| $ is the identity map on objects, and $H_{ X, Y } (f): I \to \coprod_{f: X \to Y}$ is the canonical coprojection $\epsilon_f $.

We suppose that ${}^{c}\textbf{V}$ is a $\textbf{V}$-complete (exist limits, and are preserved by interior hom-functors $[V,-]$). Then given two small $\textbf{V}$-categories $\mathscr{A}$, $\mathscr{B}$ is defined the $\textbf{V}$-category $[\mathscr{A}, \mathscr{B}]$ where its objects are the $\textbf{V}$-functors $F: \mathscr{A} \to \mathscr{B}$ and with $[F, G]:= \int_A [F(A), G(A)] $.

(end premises)

Let $\mathscr{A}$ small, I define the $\textbf{V}$-functor $Lim: [\mathscr{A}, {}^{c}\textbf{V}] \to {}^{c}\textbf{V}$ as follow (sketch):

for $F: \mathscr{A} \to {}^{c}\textbf{V}$ let $Lim (F):= \varprojlim_{A\in \mathscr{A} } F(A)= \varprojlim |F|$ and $Lim_{F, G}: \int_X[F(X), G(X)] \to [\varprojlim_A F(A), \varprojlim_B G(B)] \cong \varprojlim_B [\varprojlim_A F(A), G(B)]$ induced by $\int_X[F(X), G(X)] \to [F(B), G(B)] \xrightarrow{[\pi_B, 1 ]} \varprojlim_A [F(A), G(B)]$.

I ask if there exist necessary and sufficient conditions (on $\textbf{V}$ or $\mathscr{A}$) for the existence of a $\textbf{V}$-left-adjoint of $Lim$.

I seems that the hypothesis $\textbf{V}\cong L_* (|\textbf{V}|)$ and (then) $\mathscr{A}\cong L_* (|\mathscr{A}|)$ is enough, but this condition is very rough...

PS. this question is induced by pag.14.2 of the article Closed Categories, lax limits and Homotopy limits, J. W. Gray (JPAA 19 (1980) 127-158)

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