[Irish sources and comments are below the English]

Copulation

(Some notes about the copula)

From the Official Standard (An Caighdeán Oifigiúil):

Is éan é an faoileán. [the seagull is a bird (seagulls are birds)]

So when there are two nouns in a classification sentence, the 2nd one being definite, there is a pronoun used. So this is thesentence from our lesson (Progress in Irish chapter 33): Is múinteoir é Séamas. [Séamas is a teacher].
But, in the Christian Brothers' Grammar (Graimméar Gaeilge na mBraithre Críostaí), we get:

Is éan smólach (indefinite noun). [A thrush is a bird] An eitleoir Tomás? [Is Tom a flyer?] (is múinteoir Séamas!), without the pronoun.
So either way is acceptible in classification sentences. Raymond reads it without the pronoun on our recording. I prefer the pronoun... The Christian Brothers explicitly say that SOMETIMES the pronoun is used:

"Uaireanta cuirtear forainm pearsanta i gcomhaisnéis le hainmní ainmfhoclach cinnte."
[Sometimes a personal prounoun is put in apposition (stuck together with) a definite noun subject (comhaisnéis is really comh + fhaisnéis - together information(together predicate)) ]

Fo-ainmní a thugtar air scaití [It is called a sub-subject at times]: Is áit dheas é Dún Garbhán [Dun Garvan is a nice place]; Nach peileadóir maith é Seán? [Isn't Sean a good soccer player?]

This is a big subject, and eventually you'll learn all the little niceties. But here are some technical details which might be good to study:

An abairt aicme - the classification sentence:

Seo gnáthord na bhfocal: copail + faisnéis + ainmní

[Here's the usual order of the words: copula (is) + information (predicate) + subject]
So classification sentence: Is + múinteoir (é) + Séamas

An abairt ionannais - the indentification sentence:
Copail + faisnéis + ainmní (same usual order)
Here's the real explanation for why that 2nd pronoun crops up:
"Ní féidir an t-alt ná ainmfhocal a chur go díreach ar lorg na copaile in abairt ionannais. Chun

é sin a sheachaint is iondúil forainm pearsanta oiriúnach ar a dtugtar an fhofhaisnéis a chur

ar lorg na copaile."
[It is not possible to put the article or a noun directly following the copula in an identification sentece. To avoid that, it is usual to put a suitable personal pronoun, called the sub-predicate, following the copula.]
So the one that comes first is actually the "extra" one.
Is + é (can't put a noun here!) Séamas + é.

Another useful fact:

"Is iondúil gnáthfhoirm an fhorainm phearsanta a úsáid mar ainmní, agus ceann de na foirmeacha treise mar fhaisnéis..."
[It is usual the oridnary form of the personal pronoun to use as a subject, and one of the emphatic forms as predicate..."]
Is mise an múinteoir (copula + predicate + subject). Is múinteoir mé (same order).

I love how often the grammar books hedge their bets by saying "It's usual"... And those dot dot dots I put are for your protection...

It is possible to switch the subject and predicate in the identification sentence - usually sticking 'ná' before the subject...But now we're getting fancy!

copail (+ fofhaisnéis) + ainmní (+ ná) + faisnéis

is + é + an peaca a rinne sé (+ ná) + fear a mharú
[The sin that he did was to kill a man.]

Here's another related topic (for chapter 34 in PII particularly). I always try to use 'ní ba' in the past tense, but it is not technically necessary - you can use 'níos' for all tenses. But I prefer to distinguish... (this also talks about the past habitual and the conditional, which can also use 'ní ba')

[from An Caighdean Oifigiúil]

(f) dobhriathar breischéime, e.g., níos, ní ba:

Tá Seán níos airde ná Máire. [Seán is taller than Mary].

Bhí siad ní ba ghile ná an sneachta. [They were brighter than snow].

Tabhair faoi deara gur féidir níos a úsáid sna haimsirí go léir ach nach n-úsáidtear ní b’/ní ba ach san aimsir chaite, san aimsir ghnáthchaite agus sa mhodh coinníollach. Déantar ní b’ a úsáid roimh aidiacht dar tús guta nó fh- + guta, e.g.,

[Note that it is possible to use níos in all the tenses but ní b' / ní ba is only used in the past tense, the habitual past and the conditional. One uses ní b' before a vowel or fh, e.g.]

Ba ghá dóibh dréimire níos fearr/ní b’fhearr a fháil anuraidh [ They needed to get abetter ladder last year]; Dá mbeinn bliain níos óige/ní b’óige, d’imreoinn an cluiche [ If I were a year younger, I would play the game];

agus déantar ní ba a úsáid roimh chonsan, e.g.,

[And one uses ní ba before a consonent, e.g.]

Bhí na braillíní níos gile/ní ba ghile ná an sneachta. [The sheets were brigher than the snow].

The follwing is another section from the 1961 Teach Yourself Irish (which is Munster Irish - a few things are not in other dialects, like 'so, san' instead of 'seo, sin'). It's supplemental to our own lessons, but might help clarify some things. They even say that the sentence structure for these is complicated... So more examples and practice can help!

The Copula

In the simple sentences of Lesson I you have learned the verb tá which means “is” in such phrases as “the hat is on the table” or “the day is cold, the bag is empty,” etc. But in sentences of definition or identity, you must use a different verb, namely is (rhymes with Eng. “hiss”), which is called the copula, because it merely joins two notions.[For those who know Spanish, the distinction between ser and estar will be helpful.] The forms are easy, as this verb has no persons or number and only two tenses, present is and past (and conditional) ba; but the syntax of these sentences is troublesome and requires careful study and practice.
Definition is to say what a person or thing is: “it is a book, a horse, a hill”, etc. A sentence of identity says who or which he, she or it is: “it is my book, his horse”; “he is John Smith”. And these two types have slightly different constructions. You have seen that é, í, iad are the forms of the pronoun as object of a transitive verb. They are also the forms used as subject of the copula.
(a) The sentence of definition is then is leabhar é “it is a book”, is cloch í “it is a stone”, is fir iad “they are men” (verb-predicate-subject); or, with the demonstratives so, seo “this” and san, sin “that”: is leabhar é seo “this is a book”, is cloch í sin “that is a stone”, is fir iad san “those are men”.
(b) In the sentence of identity the pronoun occurs twice, before and after the noun, unless the subject is the demonstrative. Suppose you want to say simply “it is the book” (not the paper or the pen), you must say is é an leabhar é;
“they are the boys” is iad na garsúin iad; “it is John” is é Seán é. But the demonstrative is not repeated after a definite noun: is é sin an leabhar “that is the book”, is iad so na garsúin “these are the boys”, is é sin Seán “that is John”. If both the subject and predicate are definite nouns, e.g. “Tom is the old man”, “John is my son”, “the big book is the prize”, then the order is: is é Tomás an sean-duine; is é Seán mo mhac; is é an leabhar mór an duais. The notion to be emphasised precedes, but a definite noun may not follow the copula directly.
(c) The first type, however, has a common alternative form with the old neuter pronoun ea: fear is ea é “it is a man”; bó mhaith dob ea í sin “that was a good cow”. The past in this position is dob (the particle do and b’). And the type is é sin an fear has an alternative form sin é an fear, which is that commonly used. [Standard Irish is Bó mhaith ab ea í]
These four types should be memorised:
is fear é or fear is ea é “it is a man”
is é an fear é “it is the man”
is é sin an fear or sin é an fear “that is the man” [or Sin an fear, leaving of the 'is é' at the beginning]
is é Seán mo mhac “John is my son”
(d) In such a sentence as “good advice is a great help”, the normal construction is permissible: is cúnamh mór comhairle mhaith; but when there is emphasis on the predicate, the common usage is to make the adjective predicative and the noun definite, so that the logical subject appears in apposition: is mór an cúnamh comhairle mhaith. Thus the common form is: is maith an rud é “it is a good thing”, rather than rud maith is ea é. This form gives greater emphasis than (c). This form does not occur in Northern Irish, but tá with the preposition i n- may be used instead: (is) buachaill maith atá ann “he is a good boy”, (is) bó mhaith do bhí inti “she was a good cow”. This latter idiom appears in Southern Irish only in the negative form: níl ann ach cleasaí “he is only a trickster”, níl ann ach leath-scéal “it is only an excuse”
Learn the following three sentences by heart:
is breá an lá é “it is a fine day”
is mór an trua é “it is a great pity”
is láidir na fir iad “they are strong men”.
For emphasis a noun-subject may here be preceded by a pronoun: is maith an múinteoir é Seamas “James is a good teacher”.