  We give an alternative treatment of the foundations of parametrized spectra,
with an eye toward applications in fixed-point theory. We cover most of the
central results from the book of May and Sigurdsson, sometimes with weaker
hypotheses, and give a new construction of the bicategory $\mathcal Ex$ of
parametrized spectra. We also give a careful account of coherence results at
the level of homotopy categories.
  This is an expository work, akin to a set of lecture notes, encompassing and
extending the more novel material in the paper "A convenient category of
parametrized spectra." Our primary goal is to give careful and explicit
explanations for the standard facts surrounding the bicategory of parametrized
spectra and the Reidemeister trace. In particular, our proof of
Costenoble-Waner duality gives a more explicit account of how the concrete
index formulas of Dold are related to the abstract theory of bicategorical
duality from May and Sigurdsson, Ponto, and Shulman.
