Let $\Lambda$ and $\mathcal{C}$ be categories and $\mathcal{W}$ a subset of morphisms of $\mathcal C$ such that there exists the localization $L: \mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$. Denote $$ \mathcal{W}^{\Lambda} = \{\theta \in \mathrm{Mor}(\mathcal{C}^{\Lambda}) \,|\, \theta_\lambda \in \mathcal{W}, \forall \lambda \in \Lambda \}$$
The functor $L_*: \mathcal{C}^{\Lambda} \to \mathcal{C}[\mathcal{W}^{-1}]^{\Lambda}$ satisfies that $F(\theta)$ is an isomorphism for all $\theta \in \mathcal{W}^{\Lambda}$. In what conditions over $\mathcal{W}$ or $\Lambda$ this is the localization $\mathcal{C}[\mathcal{W}^{-1}]^{\Lambda} \simeq \mathcal{C}^{\Lambda}[(\mathcal{W}^{\Lambda})^{-1}] $?
Maybe if $\mathcal{W} \supseteq \{F(\lambda \to \lambda') \,|\, F \in \mathcal{C}^{\Lambda}, \lambda \to \lambda' \text{ morphism in } \Lambda\}$ ?
I would like to use it to see when localization $L$ preserves $\Lambda-$limits.