Should the natural homomorphism of field extensions $i: k \to F$ in this proposition be written as $k \to k[x] \to F$ or $k \to k[t] \to F$?

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From Algebra: Chapter $0$ by Aluffi:

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Since $F=k[t]/(f(t))$, is the composition in the last line correct?

Wouldn't it make more sense to write $k \to k[t] \to F$?