For the sphere fails to be path-connected as it consists of two disjoint points, hence for any there exists only one loop with that basepoint, thus .

For we regard a continuous loop as an endomorphism . We claim that Degree of a circle endomorphism constitutes an isomorphism

This is well-defined and injective since Circle endomorphisms are homotopic iff they are of equal degree, and it is surjective because has degree . Let be paths with base and let be the required continuous functions so that the following diagram commutes in for :

then the corresponding lift for the concatenated path is given by

and hence . Hence is an isomorphism, so .

For see Seifert-Van Kampen-Brown theorem.