A diagram in category theory is said to commute when for all objects $A$ and $B$ in it, every the composite morphism resulting from a possible path from $A$ to $B$ are the identical. Does that mean that if the following diagram commutes,
Then $g = h \circ e = h \circ e \circ (f \circ e) = h \circ e \circ (f \circ e)^2 = h \circ e \circ (f \circ e)^3$ and so on?
If so, then commutative diagrams with antiparallel arrows (like $e$ and $f$ in the above diagram) must be heavily restricted...