In 2-categories one can consider a notion of colimit whose universal property is expressed in terms of oplax transformations, rather than 2-natural transformations. These are called oplax colimits. Examples include the coKleisli category of a comonad and the Grothendieck construction for functors into Cat. By applying the theory of oplax-morphism classifiers one can express oplax colimits as ordinary weighted Cat-colimits for particular sorts of weights. I plan to summarise some of the work done during my phd (supervised by Richard Garner) toward describing these weights; in particular, characterising the saturation as the class of coalgebras for oplax-morphism classifiers. If time allows I might also discuss free completions of 2-categories under colimits for this class of weights.