#First we #reformulate #ordinary #colimits in the #language of #tensor #products, in a #way that #suggests #more #general #weighted colimits.

#Assume for the #moment that the #receiving category C has all #coproducts and #coequalizers. As is #well #known, it #follows that C #has #all colimits; the #proof is we can #write #down a #formula for the colimit of F:J→C: as a #coequalizer of a #pair:

∑j,k∈Ob(J)hom(j,k)×F(j) [[→ →]] ∑k∈Ob(J)F(k) [[→]] colimJF