On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachment giving Munkres pages 333-334)

"Suppose that is a continuous map that carries the point of X to the point of Y.

We denote this fact by writing:

If f is a loop in X based at , then the composite is a loop in Y based at "

I am confused as to how this works ... can someone help with the formal mechanics of this.

To illustrate my confusion, consider the following ( see my diagram and text in atttachment "Diagram ..." )

Consider a point that is mapped by f into i.e.

Then we would imagine that is mapped by into some corresponding point ( see my diagram and text in atttachment "Diagram ..." )

i.e.

BUT

But (see above) we only know of h that it maps into ? {seems to me that is not all we need to know about h???}

Can anyone please clarify this situation - preferably formally and explicitly?

Peter