To every topological space "X" with distinguished point "x"0, one can define the fundamental group based at "x"0, denoted . This is the group of homotopy classes of loops based at "x"0. If is a morphism of pointed spaces, then every loop in "X" with base point "x"0 can be composed with "f" to yield a loop in "Y" with base point "y"0. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from to . We thus obtain a functor from the category of pointed topological spaces to the category of groups.