Let $\mathcal{C}$ be a small 1-category and let $\mathcal{M}$ be a category enriched over the presheaf category $\widehat{\mathcal{C}}$ which is complete as $\widehat{\mathcal{C}}$ enriched category. Denote by $\Gamma = \mathrm{Hom}(1, -) : \widehat{\mathcal{C}} \rightarrow \mathrm{Set}$ the global sections functor, by $\mathrm{Lim}(\mathcal{M}, \widehat{\mathcal{C}})$ the 1-category of enriched limit-preserving functors and their enriched natural transformations and by $\mathrm{Lim}(\mathcal{M}_0, \mathrm{Set})$ the limit preserving functors and natural transformations from the underlying 1-category $\mathcal{M}_0$ to $\mathrm{Set}$. Composition with $\Gamma$ induces a functor $\Gamma \circ - : \mathrm{Lim}(\mathcal{M}, \widehat{\mathcal{C}}) \rightarrow \mathrm{Lim}(\mathcal{M}_0, \mathrm{Set})$. Is it true that this is an equivalence of 1-categories? I think this holds because of the isomorphisms $$ \mathrm{Hom}(\mathbf{y}(c), G(X)) \cong \mathrm{Hom}(1 \times \mathbf{y}(c), G(X)) \cong \mathrm{Hom}(1, G(X)^{\mathbf{y}(c)}) \cong \Gamma(G(X^{\mathbf{y}(c)})) $$ for all enriched limit-preserving functors $G : \mathcal{M} \rightarrow \widehat{\mathcal{C}}$, objects $c \in \mathcal{C}$ and objects $X \in \mathcal{M}$. Here $\mathbf{y}(c)$ denotes the representable presheaf, while $G(X)^{\mathbf{y}(c)}$ and $X^{\mathbf{y}(c)}$ refer to the powers in $\widehat{\mathcal{C}}$ and $\mathcal{M}$. This isomorphism implies that enriched limit preserving functors $\mathcal{M} \rightarrow \widehat{\mathcal{C}}$ are determined by the points of their images, and conversely it can be used to define a presheaf $G(X)$ given just naturally varying values for all $\Gamma(G(Y))$.
This feels like a fact that should be either false or well-known. I haven't seen this anywhere though, so I'm suspicious. Can somebody confirm that it is true (or false), and perhaps point me to a source?
