TITLE: Basic questions on the homotopy category
QUESTION [13 upvotes]: I apologize in advance if this the answer to this question is standard or well-known. I am not in any way an algebraic topologist.
$\newcommand{\s}{\mathscr}$Let $\s T$ be the category of topological spaces, $\s T_*$ the category of pointed topological spaces. We can construct the homotopy category $\s H$ and the "pointed homotopy category" $\s H_{*}$ by letting $\hom_{\s H}(X,Y)$ be $\hom_{\s T}(X,Y)$ modulo homotopy and $\hom_{\s H_*}(X,Y)$ be $\hom_{\s T_*}(X,Y)$ modulo homotopies preserving basepoints. There are obvious functors $F:\s T\to \s H$, $F_{*}:\s T_{*}\to \s H_{*}$. 
My questions are pretty basic: do $F,F_*$ preserve products? arbitrary limits? only arbitrary finite limits? coproducts? arbitrary colimits? It not, how does one compute colimits in $\s H$ and $\s H_{*}$?
If this can be found in a standard reference, just write an answer saying "this is a stupid question - this is all worked out in...," and I'll upvote and accept.

REPLY [4 votes]: A very concrete example of a cospan having no pullback in the homotopy category, which does not require any knowledge of cohomology or Moore spaces, can be found here.  It's phrased in terms of the homotopy category of groupoids, but can easily be translated to speak about $K(\pi,1)$s in the homotopy category of spaces instead.