I'm looking for a reference for the following extension property: A functor $F: C \to D $ such that for any arrow $f: x \to y$ in $C$ there is an extension $f = \hat{f} \circ i$ with $i:x \hookrightarrow z$ monic and $F\hat{f}$ an isomorphism in $D$.
My motivation is that if $D$ is an ordinal category then this is a really simple (and restrictive) definition of dimension that fits other needs of my research.
EDIT: Expanding on the dimension part: the point to use this as a definition of dimension has several advantages:
The dimension of pullbacks/pushouts acts as expected.
There are no morphisms from a higher dimension object to a lower dimension one.
This means that the dimension property is, in some sense, external to the underlying categorial structure. If we view this dimension structure as geometrical - it helps separate geometric constructions from the underlying building blocks of the construction - the category itself (this also creates difficulties modeling this dimension functor in concrete cases, but that is a technical issue in my point of view).
The last point can be extended further: The existence of such functor can be used to restrict the behavior of the morphisms related to this "external structure" (for ordinal $D$ this is point 2). The extension property is there to connect the "external structure" to the underlying category, but in a very controlled way. Now, dimension is not necessarily the only thing we want to control.
For example we can take $D=\alpha\times E$ with the first coordinate meaning dimension, and the second a discrete category - this is for any invariant under existence of morphisms (say Euler characteristic). The existence of a functor, both restricts what morphisms are allowed - only increasing the dimension and preserving some strong invariant. Every morphism can be studied as an embedding into the $D$-isomorphism class of it's codomain.