The oplax limit of an enriched category

We show that 2-categories of the form $\mathcal{B}-\mathbf{C}\mathbf{a}\mathbf{t}$ are closed under slicing, provided that we allow $\mathcal{B}$ to range over bicategories (rather than, say, monoidal categories). That is, for any $\mathcal{B}$-category $\mathbb{X}$, we define a bicategory $\mathcal{B}\mathrm{/}\mathbb{X}$ such that $\mathcal{B}-\mathbf{C}\mathbf{a}\mathbf{t}\mathrm{/}\mathbb{X}\cong (\mathcal{B}\mathrm{/}\mathbb{X})-\mathbf{C}\mathbf{a}\mathbf{t}$. The bicategory $\mathcal{B}\mathrm{/}\mathbb{X}$ is characterized as the oplax limit of $\mathbb{X}$, regarded as a lax functor from a chaotic category to $\mathcal{B}$, in the 2-category $\mathbf{B}\mathbf{I}\mathbf{C}\mathbf{A}\mathbf{T}$ of bicategories, lax functors and icons. We prove this conceptually, through limit-preservation properties of the 2-functor $\mathbf{B}\mathbf{I}\mathbf{C}\mathbf{A}\mathbf{T}\to 2-\mathbf{C}\mathbf{A}\mathbf{T}$ which maps each bicategory $\mathcal{B}$ to the 2-category $\mathcal{B}-\mathbf{C}\mathbf{a}\mathbf{t}$. When $\mathcal{B}$ satisfies a mild local completeness condition, we also show that the isomorphism $\mathcal{B}-\mathbf{C}\mathbf{a}\mathbf{t}\mathrm{/}\mathbb{X}\cong (\mathcal{B}\mathrm{/}\mathbb{X})-\mathbf{C}\mathbf{a}\mathbf{t}$ restricts to a correspondence between fibrations in $\mathcal{B}-\mathbf{C}\mathbf{a}\mathbf{t}$ over $\mathbb{X}$ on the one hand, and $\mathcal{B}\mathrm{/}\mathbb{X}$-categories admitting certain powers on the other.