Three questions on spectra To fix the notation, a spectrum is a sequence of spaces indexed over $\mathbb{N}$ (!) with structure maps as usual.
My first questions is: Does the suspension spectrum functor embed the category of spaces into spectra? It is clearly a faithful functor and injective on objects but I doubt that it is full. 
There is a unstable model structure on spectra by defining weak equivalences and fibrations degreewise.
My second question is: Does the suspension spectrum functor embed the homotopy category of spaces into the unstable homotopy category of spectra?
Define two endofunctors $s$ and $t$ on the category of spectra by setting $sX_n=X_{n+1}$ and $tX_n=X_{n-1}$ where $X_{-1}=*$. Then $t$ is left adjoint to $s$ and this is a Quillen adjunction with respect to the unstable model structure.
My third question is: Is it true that $s$ is left adjoint to $t$, too? If yes, is this also a Quillen adjunction? I have checked it and it seems to be right but somehow I feel not well about it.
 A: The answer to (1) is that the suspension-spectrum functor from based spaces to spectra is fully faithful.  Given spaces X and Y, a map between their suspension spectra $\Sigma^\infty X \to \Sigma^\infty Y$ is a collection of maps $S^n \wedge X \to S^n \wedge Y$ respecting the structure maps.  In particular, taking $n=0$ we recover the map $X \to Y$, and since the structure maps are all isomorphisms any map on level zero extends uniquely to the rest of the spectrum.  Another way to see this is that the suspension spectrum functor is left adjoint to the functor $U$ that takes a spectrum $\{E_n\}$ to $E_0$.
The answer to (2) is that the suspension spectrum functor is faithful; it has a left inverse, which is obtained from pushing the functor $U$ down to the homotopy categories.
The answer to (3) is that s is not left adjoint to t on the level of spectra  For instance, one has $s(\Sigma^\infty X) \cong \Sigma^\infty(S^1 \wedge X)$ for any $X$, and so by the adjunction mentioned in (1), we have:
$$
Hom(s \Sigma^\infty X, \Sigma^\infty Y) = Hom(S^1 \wedge X, (\Sigma^\infty Y)_0) = Hom(X,Y)
$$
However, we also have:
$$
Hom(\Sigma^\infty X, t \Sigma^\infty Y) = Hom(X,(t\Sigma^\infty Y)_0) = *
$$
