TITLE: Brown representability in slice category
QUESTION [7 upvotes]: Brown's representability theorem gives us a very nice set of conditions to check that a (contravariant) functor $Hot^{op}\rightarrow Set$ is representable. Choose an object $X$ in $Hot$. Then it is seems natural to ask whether or not an analogue of the Brown representability theorem is true for the slice category, i.e. if there exists a nice set of conditions to check whether or not a contravariant functor
$$F:(Hot/_X)^{op} \rightarrow Set$$
is representable. Is there such an analogue? I've searched online, but couldn't find an comments on the subject. My hope would be that this functor is representable if $F$ respects coproducts and sends homotopy pushouts to weak pullbacks.
One way I would hope to recover such a criterion is to consider the functor $$Hot\rightarrow Hot/_X, Y\mapsto (Y\rightarrow \{*\})$$
which by composition gives us a functor $Hot^{op}\rightarrow Set$, for which we know we can apply Brown's Theorem. I don't know however if this is enough to test representability of the functor $F$.

REPLY [6 votes]: Edgar Brown wrote two papers on this topic: one in the Annals in 1962 that focused on the category of topological spaces, and a second one "Abstract homotopy theory"
Trans. Amer. Math. Soc. 119 (1965).  
His second paper is very axiomatic, and I believe your situation is easily checked to satisfy his properties. (Check this!)  
This paper obviously predated Quillen's work on model categories (it likely partially inspired Quillen), so of course he doesn't use that language.  I confess that I have always been a bit disturbed by the number of papers on Brown Representability written by authors who show no indication that they have ever looked at the original papers. Yes, people have written about more general versions (and less general, when they assume a triangulated category!) but I encourage folks to use their library resources to look at Brown's own work.