2-category of topological spaces with continuous maps and homotopies

Definition
The 2-category of topological spaces with continuous maps and homotopies is defined as follows:

Relation with the associated 1-category
If we forget the 2-morphisms, we get the usual category of topological spaces with continuous maps.

The homotopy category of topological spaces
The 2-category described here is very special in the sense that all its 2-morphisms are invertible. Hence, we can safely take a quotient of the 1-category modulo the equivalence relation induced by the 2-morphisms. This gives rise to the homotopy category of topological spaces, where two continuous maps $$f$$ and $$g$$ are identified if they are homotopic.

Singular chain complex
The singular chain complex can be viewed as a 2-functor from this 2-category to the 2-category of chain complexes with chain maps and chain homotopies. This gives a strong form of the homotopy-invariance theorem for singular homology.

Underlying space of a cellular space
The underlying space functor can be viewed as a 2-functor from the 2-category of cellular spaces with cellular maps and homotopies to this 2-category, which sends each cellular space to its underlying topological space, each cellular map to its underlying continuous map, and each homotopy to the corresponding homotopy.

This in particular gives a 2-functor from the 2-category of CW-complexes with cellular maps and homotopies.