I have recently been trying to learn about homotopy colimits. In so doing, I have gone through way too many papers, and now I can't find the one which introduced homotopy colimits via a coend computation.
Note that I am interested in the "local" formulation (to use the terminology of Shulman's paper), as I really want to perform very concrete computations. Furthermore, I am most interested in purely categorical formulations, as the background computations all happen in type theory, not topological spaces.
Part of the motivation is to understand the paper Ring completion of rig categories, and unravel the construction given there in simpler terms.