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Equivalence classes in codomain

Let’s suppose we have a function $f:X\rightarrow Y$ and let’s define the function $g:Z\rightarrow X$ by $g=f\circ h$. I saw in my notes that the equivalence class of $g$ is a representative of the equivalence class of $f$ and it is constructed by taking a class $[x]\in Z$ such that $f(x)=g([x])$.
Is the converse true? I mean, is the following statement true:

If $f(x)=g([x])$ for some $g:[x]\rightarrow Y$, then the equivalence class of $f$ is a representative of the equivalence class of $

https://colab.research.google.com/drive/13DT00RMc3Ov06fiAURHXSWbkEK2G38rT
https://ello.co/chaigueschafre/post/qxtxxyznc6gn5e9se-wr6a
https://colab.research.google.com/drive/1sTwTFBbJ9oRe1AfE5mN71wHd-yDbK1iV
https://ello.co/8liname_pa/post/fk01fcrh6dpjotgq9yclba
https://documenter.getpostman.com/view/21833004/UzdwWmyj
https://ello.co/formigriazu/post/5rhvq0rnp1lkj3owbbtazq
https://colab.research.google.com/drive/1rjNdrvf5kGN5BdN-RwhVD-tDCg2K-paQ
https://ello.co/subnessposor/post/1xjhdg525u6rmfjeohwyva
https://colab.research.google.com/drive/14NT93otcjquxD67WODaQnLHCUSmEzOtq
https://colab.research.google.com/drive/1ux8pqGs_fYALO464iGPSaYGRjcWbsyW8