Given a filtered vector space (or module over a ring) 0=V₀⊆V₁⊆...⊆V, you can construct the associated graded vector space gr(V)=⊕_iV_i+1/V_i. Does gr(V) satisfy a universal property? What is it?

Before anybody hastily says, "it's the universal graded vector space with a filtered map from V," let me point out that it's not so simple. A map of filtered vector spaces is a map of vector spaces which respects the filtration. It's clear what the map V_i+1→V_i+1/V_i should be, but what would the map ∪V_i→⊕_iV_i+1/V_i be?