(1) *anyone who knows that scene
(2) * the "literal point" of the scene, however, is to impart humour and good cheer, not to teach Latin grammar.
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Quote from undisclosed source: "Didya hear the one about the Latin teacher who dreamed he was teaching? Then he woke up and he was!"
He pulls his ear and makes him conjugate the verb correctly. It's probably a riff on the way Latin was taught in England to the authors of the sketch. They like to poke fun at the absurdities of the English school system, as is the case with the teacher who goes on about the rule for how very simple it is to put the clothes specifically on the lower peg etc. https://www.youtube.com/watch?v=XIuC--xXMXI
I'm still a learner of latin so I'm not 100% sure but, 'Domo' is the ablative (from the house; preposition: indicating movement away from the house) while 'domum' is the accusative (to the house; direct object: the house is receiving the action of being gone to).
Here's a table with every inflection of domus. https://en.wiktionary.org/wiki/domus#Latin
Domi means "AT home" (genitive case). Domum means "TO home" (accusative case). I find it odd that the verb "to go" would use accusative case, but that appears to be what is happening here. You can see the declension here: https://www.online-latin-dictionary.com/latin-dictionary-flexion.php?parola=Domum
It's typically called 'accusative place to/into which' and is used for space and time phrases such as Romam iit, 'he went to Rome'; we are sending Paul to Rome = Paulum Romam mittimus. In general, accusative is motion towards in Latin, as in ancient Greek. In the famous Monty Python scene, domum is accusative, Romani ite domum. https://www.youtube.com/watch?v=IIAdHEwiAy8
That male dubber irritates me sometimes when he pronounce the double vowels detatched from one another. For an example "eunt" it's more simple, natural and beautiful when the "e" blends with the "u". What instead the dubber does is pronounce the vowels sharply separated, like "e - u". Please change, this is misleading for non-latin-derived language speakers.