It’s probably not correct.
The probability that in the 1880s a man travels across the state of Illinois from the village of Altona to the city of Joliet is minimal. It is possible? Yes. But any court of record could naturalize in 1880 in the United States. An immigrant would not have had to travel that far in order to complete his citizenship. There was no requirement that the naturalization process had to be completed in the same place where it was started and there’s no evidence that the person of interest originally settled in Joliet.
And even if he had, it did not matter.
The logical place for an individual living in Altona, Illinois, to naturalize in the 1880s would have been the county seat of Galesburg. Cambridge, the county seat of nearby Henry county was another nearby option.
The name of the individual, Carl Nelson, is not uncommon enough to where “the name’s the same” would be strong enough reason to assume that the person “had” to be the same.
And “had to be the same” is a phrase that can get researchers in trouble.
If I think that a person travelled 180 miles to naturalize when any court of record could do the procedure (and were doing the procedure regularly), I better have a good explanation as to why someone went that far.