The map π:X→Y{\displaystyle \pi :X\to Y} is a surjection is a surjection. In the diagram on the left, there might not exist any function f:Y→Z{\displaystyle f:Y\to Z} that when added to the (left) diagram, will make it commute; that is, there might not exist any f:Y→Z{\displaystyle f:Y\to Z} such that F=f∘π{\displaystyle F=f\circ \pi } on X.{\displaystyle X.} This motivates defining the set DF,{\displaystyle D_{F},} which is defined as the largest subset of Y{\displaystyle Y} on which it is possible to unambiguously define a function f:DF→Z{\displaystyle f:D_{F}\to Z} that satisfies F=f∘π{\displaystyle F=f\circ \pi } on π−1(DF).{\displaystyle \pi ^{-1}\left(D_{F}\right).} 
In the diagram on the right, the arrows labelled π{\displaystyle \pi } and F{\displaystyle F} are actually the restrictions to π−1(DF){\displaystyle \pi ^{-1}\left(D_{F}\right)} of the original maps π:X→Y{\displaystyle \pi :X\to Y} and F:X→Z.{\displaystyle F:X\to Z.} That is, it is the largest subset of Y{\displaystyle Y} on which a edge may be added to the diagram on the left to turn it into the triangular diagram on the right (after restricting the functions to the appropriate subsets).
Own work
Mgkrupa