presheaf of a topological basis

Let $X$ be a topological space and let $ℬ$ be a basis of its topology. We can regard $ℬ$ as a categorywith objects being the open sets in $ℬ$ and arrows/morphisms between $U,V\in ℬ$ to exists only if $U\subset V$, andwhere the only element of $ℬ(U,V)$ is the injection map $U\hookrightarrow V$. Let now $𝒞$ be a completecategory, we now define thepresheaf of $\mathrm{C}$-objects over the basis $\mathrm{B}$ of the topology of $X$ to be a contravariant functor

$$𝒫:ℬ\to 𝒞$$