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This was proven true by Ford, Luca and Pomerance in 2010. We ask the analogous question of whether infinitely often we have $\\varphi(F) = \\sigma(G)$ where $F$ and $G$ are polynomials over some finite field $\\mathbb{F}_q$. We find that when $q\\not=2$ or $3$, then this can only trivially happen when $F=G=1$. Moreover, we give a complete characterisation of the solutions in the case $q=2$ or $3$. 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