Everything stated here occurs in a category where every map is a continuous map. Suppose $C$ represents circle, $D$ represents Disc, and $P$ represents a Plane. Consider following maps:
- Inclusion of circle into disc, $j:C\rightarrow D$
- Inclusion of disc into plane, $i:D\rightarrow P$
Following is an observation:
$$\forall f, g: D\rightarrow P,$$ $(f\circ j = i\circ j) \ and \ (\exists o:C\rightarrow D \ s.t. \ g\circ j=i\circ o) \Rightarrow (\exists d:1\rightarrow D \ s.t. \ f\circ d= g\circ d) $.
Author has asked to test the above observation by constructing a counter-example. But so far, I have failed to do so. I have thought of it as concentric discs in plane, and then points inside the disc 1 move towards point $p1$ and points of disc 2 move towards point $p2$ (Imagine stretching a two dimensional rubber while standing at $p2$ perhaps via magnetism or something). But it still appears that, some points which are labelled the same by maps $f$ and $g$ move to the same location.
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