Question on universal arrows

Greetings to all the readers. I am studying category theory on my own and I did not succeed in finding an answer to the following question. Colimits can be defined in terms of universal arrows: the colimit of a functor F: S -> C is the universal arrow, in the category C^S, from F to the diagonal functor D: C -> C^S. Can we also define universal arrows in terms of colimits? If yes, how? Thank you very much Pietro Braione