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Path components

(See attached file for full problem description with proper symbols and equations)

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Let X be a topological space. Mapping a point to the path component which contains x establishes a map .
Show that for any continuous map between topological spaces, there exists a map such that the following holds:
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? for two continuous maps and we have
? for the identity we have where the latter map denotes the identity on .
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Proof:

Look at the above figure. Since is a continuous map, then maps a path connected component to a path connected ...

Solution Summary

This solution is comprised of a detailed explanation to show that for any continuous map between topological spaces, there exists a map such that the following holds:
?
? for two continuous maps and we have
? for the identity we have where the latter map denotes the identity on .

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