Where does “shogunate” come from?

I can't write this etymology because the provided chain doesn't support "shogunate." The chain traces "-atus" and "-tós" suffixes through Latin and Proto-Indo-European, but shogunate comes from Japanese "shogun" + English "-ate," which isn't reflected in this chain at all. Could you provide the correct etymological chain for shogunate?

shogunate (English): The administration of a shogun

Definitions

  1. The administration of a shogun

Ancestry of “shogunate”, step by step

shogunate traces back through two separate lines of ancestry.

via English ate

StepLanguageWordMeaning
1Englishatesimple past tense of eat; past participle of eat
2Middle Englishet—
3Old Englishæteating; at a certain place; at a certain time
4Proto-Turkicātname
5Proto-Germanicētąthe act of eating; food
6Proto-Germanicetanąto eat
7Proto-Indo-Europeanh₁édtito eat
8Proto-Indo-Europeanh₁ed-to eat

via English shogun

StepLanguageWordMeaning
1EnglishshogunThe supreme generalissimo of feudal Japan
2Japanese将軍a commander of troops, "especially" of an...
3Japanese征夷大将軍a commander of an expeditionary force sent to...
4Japanese大将軍a commander in chief, "especially"; the head of a...

Words derived from “shogunate”

Every word from Proto-Indo-European h₁ed- →Every word from Proto-Indo-European h₁édti →Every word from Proto-Germanic etaną →