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What is the conventional symbol used to express that two functions (that have the same domain and the same codomain) are the same? Is the normal equality symbol used (=), or something else, like the equivalence symbol ($\equiv$)?

As an example; would you write:

$\phi: GL_2 \rightarrow GL_2,\ \ \psi: GL_2 \rightarrow GL_2, \ \ \phi \equiv \psi$

or:

$\phi: GL_2 \rightarrow GL_2,\ \ \psi: GL_2 \rightarrow GL_2, \ \ \phi = \psi$

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If you only use the function names, then $\phi=\psi$ ought to be enough. If you instead supply an argument (an element $x\in GL_2$) and write $\phi(x)=\psi(x)$, I would say that you ought to write either "$\phi(x)=\psi(x)$ for all $x$" or "$\phi(x)\equiv \psi(x)$".

However, I don't think there are any strict conventions here. Any way you can make it clear that it's an equality and not an equation should be good enough.

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If they agree on all elements, then you usually put equality $=$. Remember that a function can be seen as a vector of elements (at every coordinate you have the corresponding value), so agreement on all coordinates means equality as vectors.

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