$\newcommand{\B}{\mathcal{B}}$ $\newcommand{\C}{\mathcal{C}}$ $\newcommand{\hom}{\mathrm{Hom}}$ There is a standard notion of a category $\C$ enriched over another category $\B,$ which informally captures the intuition that, for any pair of objects $X,Y\in \C,$ the mapping space $\hom(X,Y)$ is "enriched" to an object of $\mathcal{B}.$ For example, the category $\mathcal{C} = \mathrm{ComplexMfd}$ of complex manifolds is enriched over the category $\B = \mathrm{Top}$ of topological spaces, since the set of holomorphic maps $X\to Y$ between complex manifolds is a topological space in the compact-open topology.
I want to use a similar term to "$\C$ is enriched over $\B$" for a more general situation, where instead of asking that $\hom_\C(X,Y)$ is an object of $\B$ as before, I only want $\hom(X,Y)$ to be a presheaf on $\B$ (i.e., a functor from $\B$ to sets). For example if $\C= \mathrm{ComplexMfd}$ is once again the category of complex manifolds and $\B=\mathrm{Mfd}$ is the category of smooth manifolds, it is no longer the case that $\hom_\C(X,Y)$ is an object of $\B$: indeed, the space of holomorphic maps is in general not even finite-dimensional. However, for any pair of complex manifolds $X,Y$ and a real manifold $T,$ there is a notion of "a family $$\{f_t:X\to Y\mid_{t\in T}\}$$ of holomorphic maps $X\to Y$ indexed by $T$" (explicitly, these are families of maps indexed by $t\in T$ such that the associated map of sets $T\times X\to Y$ is a map of manifolds).
In this situation, $\hom(X,Y)$ is a presheaf on $\B$ (which takes $T\in \B$ to the set of families of maps $X\to Y$ indexed by $T$), and what I am looking for is equivalent to saying that $\C$ is enriched over presheaves on $\B$. But the notation "enriched over presheaves" is a mouthful. I want a better term, like "$\C$ is weakly enriched over $\B$" or something like this. Does such a term exist?
