The concept of a cospectrum is the formal dual to the concept of a (sequential) spectrum object. Where a sequential spectrum is a sequence of objects$(X_n)$ equipped with a sequence of morphisms of the form $\Sigma X_n \to X_{n+1}$ (for $\Sigma$ a reduced suspension operation), a cospectrum is such a sequence of objects, but with the direction of the structure maps reversed: $X_{n+1} \to \Sigma X_{n}$.

References

Elon L. Lima, The Spanier-Whitehead duality in new homotopy categories, Summa Brasil. Math. 4 (1959) 91–148 (1959) MR0116332

A. Jankowski, Remarks on generalized homology theories, Comment. Math. Prace Mat. 15 (1971) 185–199 MR405404

Mizuho Hikida, On CW cospectra, Hiroshima Math. J. Volume 11, Number 2 (1981), 347-368 (Euclid)

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