h = f*g
If $f$ or $g$ is not well defined, then it may happen that $f*g$ is well defined and not equal to the map $x \ \to\ f(g(x))$ but equal to the map induced by the values $f(g(gen))$ where $gen$ is a generator for $M$. Here is an example of this fact for rings.
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The source of this document is in GradedLieAlgebras/doc.m2:2635:0.