5th Pope Evaristus = Emperor Nerva. (Not sure if Evaristus matches Nerva or Trajan, but Nerva seems more likely match.)
Dates match: Pope Evaristus/Eucharistus (79 or) 97/98/99/100-105/106/107 (compiled from six different official popes lists). Emperor Nerva reigned 96-98 (+ coins for deified Nerva not issued until 10 years after his death) Both dates coincide 97-98 (+ 10 yrs after); the dates 96-98 & 97/98/99 are similar.
(Names match? Both names Nerva & Evaristus sound/look similar (both erva/Evar).)
Meaning of name match: Evaristus/Eucharistus/Aristus is from eucharia "gracious", &/or could mean eu "good" & aristo "best"? Nerva "was a member of the Italian nobility". Nerva described as "his benign nature", "mild disposition respected by the good", "well-intentioned", "the dawn of a most happy age, [when] Nerva Caesar...."
Number match: Evaristus is 5th pope from "Peter". Nerva is 5th emperor from Nero exlcuding the 3 "usurpers".
Both are in Rome.
Both are pontiffex maximus. Evaristus is bishop / "pope" / "pontiffex maximus" / "supreme pontiff" (Annuario Pontificio, Liber Pontificalis). Nerva was pontiffex maximus (the emperors from Augustus to Gratian were pontiffex maximus). [The argument that the titles pontiffex maximus and pope were not used by/for bishops of Rome until later (mid-5th & mid-6th centuries) and so is an anachronism is not necessarily valid because bishop/pope/pontiff are still related.]
Evaristus successor of Peter. Nerva "he was a textbook illustration of what nowadays is called the "Peter Principle"".
With both the popes & emperors before & after also match all in order. 1 Peter (Nero) = Nero (Peter) 2 Linus = Vespasian 3 Cletus = Titus 4 Clement 1 = Domitian (Clemens) 5 Evaristus = Nerva 6 Alexander 1 = Trajan (Alexander) 7 Sixtus 1 = Trajan? or Hadrian? 8 Telesphorus = Hadrian? 9 Hyginus (Graecvs) = Hadrianus (Graeculus) 10 Pius 1 = Antoninus Pius etc. See each separate pope & emperor chapter for matches details.
Evaristus: "Saint", shown with a halo/aura. Nerva: deified.
No contemporary proof of existence of any pope Evaristus. "Many popes in the first three centuries of the Christian era are obscure figures". Though Eusebius mentions Evaristus in ca 310. Does Eusebius say he was bishop? Even if he does, the date they first came up with the popes list is not certain, maybe they first came up with the pope/emperor correspondences even as early as Eusebius?