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. The potential audience for this work may extend outside the boundaries of modern homotopy theory, so our treatment is structured to use as little technology as possible. In particular, many of the results are stated without using model categories. We also illustrate some applications to fixed-point theory.
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