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I'm an engineering graduate student who is involved in control engineering research. I've just recently started to extract some (relatively) abstract properties from the theory that I am developing, and I'm not sure how to compare different functional compositional with each other.

As an example, let's consider the following commutative diagram:

enter image description here

The only thing that I need argue is that applying functions $f$ and $g$ to $X$ yield $Y$, regardless of the order according to which the functions are applied to $X$. Despite of the trivial nature of the question for mathematicians, I need to know that which one of the following statements is valid (or seems to be better) to compare functional compositions?

$$f \circ g = g \circ f$$ $$f \circ g \equiv g \circ f$$

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    $=$ is good, in my experience $\equiv$ is used mostly for equivalence relations, as in "these functions are equal up to a constant" for example. But something seems wrong with your commutative diagram: $g$ is both a map $X \to A$ and $B \to Y$ (and similar complaints for $f$). – Joppy Oct 22 '17 at 07:13
  • @Joppy: Basically I have two operators that can be physically applied to an input like $X$, and both the following cases return the same result, i.e., $Y$. $f(g(X)) = g(f(X)) = Y$. Can you explain why this is wrong to be commutative diagram? –  Oct 22 '17 at 07:16
  • Usually seeing $X \xrightarrow{g} A$ in a diagram means that $g$ is a function with domain $X$ and codomain $A$. – Joppy Oct 22 '17 at 07:22
  • @Joppy: Well, all of the objects in this diagram are sets, so the domains and the codomains are all the same. –  Oct 22 '17 at 07:26
  • If all the domains and codomains are the same, in a commutative diagram they should all be labelled with the same letter. But supposing we replace $A$, $B$, and $Y$ with $X$, then you would have a valid diagram, and saying "this diagram commutes" would be the same as saying $f \circ g = g \circ f$. – Joppy Oct 22 '17 at 07:29

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