§ Right Kan extensions as extending the domain of a functor (WIP)

§ First over functions (fake category fluff)

Given a function g:CEg: C \rightarrow E and an embedding j:CDj: C \rightarrow D, then the Right Kan extension of gg along jj, denoted g/jg/j is a new function g/j:DEg/j: D \rightarrow E. So we are extending the domain of gg, along the extender jj. Informally, we write the new function as g/jg/j because:
C---
|   \-g---*
j         |
|         |
v         v
D--g/j--->E
foo(j(c)) = g(c)
foo = (g/j)(c) 
g/j(j(c)) = g(c)

§ Next over preorders (real category stuff)

The kan extension provides us a bijection, for any function f: D -> E
C---
|   \-g---*
j         |
|         |
v         v
D--g/j--->E
D------f->E

hom(f.j, g) ~= hom(f, g/j)
That is, if we have some way to make congruent f.j with g, then we can "split" the congruence to have f.j congruent with (g/j).j. Cancelling j, we can have f congruent with g/j. Consider a preorder with a ordering . Equip with a monoidal structure <>, which is a monotone map with a neutral element. (For example, integers with multiplication and ). The bijection of hom-sets is equivalent to saying
m*k <= n iff m <= n/k
(how? I have no idea; I gotta work this out!)

§ Question: What happens in the context of vector spaces?

Since linear algebra is probably the nicest thing we have in math, I really want to understand what happens in the linear algebra case. I don't really understand how to make the correct version of a kan extension inside VectVect, though. A kan extension seems to fundamentally be a 2 categorical construct, than a 1 categorical construct.