I am referring to the long Proposition 1.23 of U. Schreiber's notes in nlab.
We let $X$ be a functor from $StdSphere\rightarrow Top^{*/}_{cg}$.
He states that there is a map , where $X_i^{seq}=X(S^i)$,
$$ S^1 \rightarrow Map(X_n^{Seq}, X_{n+1}^{seq})_*$$
where did this map come from?
StdSphere is enriched over the category of pointed topological spaces. $Top^{*/}_{cg}$ is also enriched over the category of pointed topological spaces. Let C and D be categories enriched over some category A. If you now recall the definition of a functor of enriched categories $F:C\rightarrow D$. One requirement is that there has to exist a morphism for all a & b objects in C.$$C(a,b)\xrightarrow F D(Fa,Fb)$$ fitting into a certain commutative diagram, for the case $A = $Set this map is just the one taking morphisms in C to morphisms in D and the commutative diagram ensures that $F(f \circ g) = F(f) \circ F(g)$.
In our case $C = StdSphere$, $D = Top^{*/}_{cg}$ and $A = Top_*$ so for a functor X of enriched categories there has to exist a map. $$S^1 = C(S^i,S^{i+1}) \xrightarrow X D(X^{seq}_i, X^{seq}_{i+1}) = Map(X^{seq}_i, X^{seq}_{i+1})$$
I hope that answers your question.