  This is the first paper of a series which aims to set up the cornerstones of
Koszul duality for operads over operadic categories. To this end we single out
additional properties of operadic categories under which the theory of
quadratic operads and their Koszulity can be developped, parallel to the
traditional one by Ginzburg and Kapranov. We then investigate how these extra
properties interact with discrete operadic (op)fibrations, which we use as a
powerful tool to construct new operadic categories from old ones. We pay
particular attention to the operadic category of graphs, giving a full
description of this category (and its variants) as an operadic category, and
proving that it satisfies all the additional properties.
  Our present work provides an answer to a question formulated in Loday's last
talk in 2012:``What encodes types of operads?''. In the second and third papers
of our series we continue Loday's program by answering his second question:
``How to construct Koszul duals to these objects?'', and proving Koszulity of
some of the most relevant operads.
