Classes of monomorphisms - a definition I have a certain construction relating to subobjects in an arbitrary category. Now the nlab article on subobjects says:

More generally, in some contexts we may take “subobject” to mean an isomorphism class of morphisms $i: a\to c$ satisfying some suitable condition other than being a monomorphism (usually a stronger one).

And this makes perfect sense, now when trying to generalize the construction to some subclass of monics (in order to find the 'correct' construction for a category when there's a better notion of subobjects), I came to realize that I needed some pseudo-functors to be in this "subclass" of monics. For example, I needed monics of presheaves to be calculated pointwise, and some 2-morphisms of Grothendieck fibrations to be 2-monic in this stronger sense.
This leads me to my question: is there a (universally accepted?) definition of "classes of monic" that behaves well in higher categories as well? I really don't want to reinvent the wheel here.
One such definition could be a small category $D$ such that $n$-monics are $n$-limits of $n$-functors from $D$. This seems like a good definition, only it misses some classes such as effective monics which are not given like this. Is there a better one?
 A: I think I found an answer:
$n+1$-categories can be viewed as categories enriched in $n$-categories, and if we had a notion of "monomorphism" for $n$-categories, whatever that notion would be for the higher version, the $hom$ bifunctor should preserve this structure (similar to the fact that fully faithful functors are the monics in $Cat$). 
So we are left with defining the enriching 2-category of monoidal categories equipped with "monomorphism" structure and this would determine a "good" definition for higher "monomorphisms".
I believe the definiton for this category should be $(\mathcal{V}, m)$ where $\mathcal{V}$ is monoidal and $m$ is a lax monoidal endofunctor $m:\mathcal{V}\to\mathcal{V}$ that is an identity on objects.
$m$ is to be thought of as the "inclusion of the subcategory of monomorphisms"
The morphisms (at least for the same $\mathcal{V}$) should be factorizations of the endofunctors which correspond to 'enlarging' the class of monomorphisms.
Now this definition allows degenerate classes, such as given by the $id$ or ones remembering only isomorphisms, so this could use some refinement.
