I believe it is true that if a functor $F:Top\to Top$ preserves weak homotopy equivalences then it is related by a natural weak equivalence to some other such functor that is in some sense continuous, perhaps continuous on morphisms in the sense that when a set map $K\to Top(X,Y)$ corresponds to a continuous map $K\times X\to Y$ then the resulting set map $K\to Map(F(X),F(Y))$ corresponds to a continuous map $K\times F(X)\to F(Y)$ when $K$ is a cell.

Question 1: Is this true? If so, what is a reference?

There are also simplicial analogues of this: instead of seeking to replace $F$ by a continuous functor, one could seek to replace it by a simplicial functor, i.e. a map of simplicially enriched categories from $Top$ to $Top$ (so that in addition to assigning an object $F(X)$ to each object $X$ it also assigns a map of ``simplicial hom-sets'' to each pair of objects). I believe this can be done by realizing the simplicial object $$n\mapsto F(X^{\Delta^n}).$$ Perhaps another way is to use the cosimplicial object $$n\mapsto F(X\times \Delta^n).$$ Here $F$ might be a functor from simplicial sets to simplicial sets rather than from spaces to spaces.

Question 2: Again, references please.

Question 3: Can a simplicially enriched replacement for $F$ be used to make a continuous replacement? I get a bit muddled here by the thought that while continuity is a property simplicial enrichment is a structure.

ADDED LATER:

To clarify: I'm quite sure that I see how to replace a functor (that preserves weak equivalences) by a weakly equivalent one that is simplicially enriched, in both the topological and the simplicial setting. I think that this is "well known", and I would be grateful for a good reference.

I would be even more interested in a method of replacing such a functor by a weakly equivalent one that is topologically enriched (in the topological setting, of course). I think that somebody told me years ago that this can be done.

And to address a possible point of confusion: Although realizations of "simplicial hom-sets" can be used as "hom-spaces", that's not quite the same as putting a topology on the ordinary hom-set.

Answers to any of this would help both me and a student of mine with things that we are writing.