= Solution
A <topological spectrum> can be described sequentially by <based spaces> $E_j$, $j\ge0$, and structure maps
$$
\sigma_j:S^1\wedge E_j\longrightarrow E_{j+1}.
$$
Iterating these maps gives compatible maps $S^{k-j}\wedge E_j\to E_k$. Equivalently, an <indexed prespectrum> on a <universe for spectra> $U$ has spaces $E(V)$ and unital associative structure maps $S^{W\ominus V}\wedge E(V)\to E(W)$ for finite-dimensional inclusions $V\subset W\subset U$. Here $S^A$ denotes the <one-point compactification> of the real <inner product space> $A$.
A <map of topological spectra> $u:D\to E$ is a family of based maps $u_j:D_j\to E_j$ satisfying
$$
u_{j+1}\sigma_j^D=\sigma_j^E(1\wedge u_j).
$$
The indexed version requires the same compatibility for every inclusion of indexing subspaces. These are point-set maps, before passing to <spectrum homotopy> or the <stable homotopy category>.
An <Omega-spectrum> is a <topological spectrum> whose adjoint structure maps
$$
E_j\longrightarrow\Omega E_{j+1}
$$
are <weak homotopy equivalences>. Consequently $\pi_kE=\pi_{k+j}E_j$ whenever $k+j\ge0$, using the structure identifications. In the genuine indexed point-set convention, the word spectrum already includes the stronger condition that $E(V)\to\Omega^{W\ominus V}E(W)$ is a homeomorphism; a general system without that condition is called an <indexed prespectrum>. Its associated spectrum is obtained by <spectrification>. The sequential and indexed conventions present the same stable theory.
A <cell spectrum> is built from the zero <topological spectrum> by successive stable cell attachments. Concretely, attach a free spectrum on the inclusion $S^{m-1}\hookrightarrow D^m$ at some level $j$ by a pushout along its boundary map; its quotient is the stable cell sphere $\Sigma^{m-j}\mathbb S$. Iterating such attachments and taking the colimit gives a <cell spectrum>, with successive quotients wedges of integer <suspensions of spectra> of the <sphere spectrum>. Negative cell degrees are allowed by choosing $j>m$. In the genuine indexed model apply <spectrification> to this cellular prespectrum. \b[The compatibility of the structure maps is part of both the objects and their maps.]
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