homotopy invariant functor

Let be a category, category Top (resp. category Top*, category TopPairs) and a functor. Then is said to be homotopy invariant, if factors through Category hTop (resp. Category hTop*, Category hTopPairs)

see:

1. homotopy invariant functor and homotopic maps
2. homotopy invariant functor and interval
3. homotopy invariant functor and homotopy equivalent spaces