# Set of fixed points of a collection of composed functions

Suppose $$h:S^nrightarrow S^n$$ is a continuous function from an $$n$$-dimensional simplex in itself. Besides, assume that the set $$Sigma$$ of its fixed points is composed by a finite number of connected components.
Now consider any collection $$G$$ of finitely many continuous functions $$g_i:S^nrightarrow S^n$$ indexed by $$iin I$$. Assume that any two functions in $$G$$ are homotopic.
Create a new collection $$tilde{G}$$ of continuous functions $$hcirc g_i$$.
Assume that any fixed point of a function $$hcirc g_i$$ is also a fixed point of $$h$$. Besides given a fixed point $$sigma$$ of $$h$$, there exist a function $$g_i$$ in $$G$$ such that $$sigma$$ is also a fixed point of $$hcirc g_i$$.
Does exist one (connected) component of $$Sigma$$ that includes at least one fixed point of every function $$hcirc g_i$$?