"It is cool in summer, and cold in winter."
For anyone wondering what's going in this sentence, this is called "continuative form used as a conjunction", in Japanese this is called 連用中止形【れん・よう・ちゅう・し・けい】
Basically, you change the predicate of the first clause into its continuative form and attach the second clause after that,
In this particular example, you are changing 涼しい into 涼しく in order to use this ～く form as a conjunction, in doing so, you can attach a different clause at the end and it this produces a full sentence with 2 parts. You can translate this ～く as "and..." in English but is better if you understand the logic of it in Japanese, imho.
So a literal translation of「夏は涼しく、冬は寒い」would be "speaking of summer, it is cool (and) speaking of winter, it is cold" (again, this is not really a good "and" equivalent but this is the only way to connect these in English). Then you add the です to make the sentence more polite at the end and we have the full sentence that Duo is showing.
You might have seen this form of adjectives being referred as "adverbial form" or "negative form", in reality, is all about connective usage,
when you wanna negate an i-adjective for example, you change it to the connective form and add ない or ありません at the end like 涼しい → 涼しく +～ない →「涼しくない」"not cool".
and when you wanna use these adjectives as an adverb you do the same to attach the verb after then like 涼しく + ～吹きます →「涼しく吹きます」"blows coolly".
There are other examples in duo using this very same grammatical point but with verbs, in that case you use the i-stem of the verb which is called "ren'youkei" duh!, and then attach another clause to the sentence. It works exactly the same but I believe those examples sound more literary and as such they mostly appear in text and in highly formal speech. In those cases the te-form is preferred.
Or is it more that using that 〜く makes the word itself a conjunction?
yeah, I guess you could say that, but is more like the second clause adheres to the first one, similar to how you join i-adjectives with ない like in 涼しく～ない "given the concept of being cool, it is not". So in that manner, it is very similar to "and" from English, but when using "and" you're also losing the written language nuance, it's something like the literary sounding texts you find in the bible (in English).
I recommend this article for more if you are interested:
In any case, most of the time you will use te-form instead, especially in speech. This is something that only appears in written language or in very very formal speech.
Thank you so much for the thorough explanation and follow-up. I'm using Duolingo after a ten year gap of learning Japanese; the class was full-immersion so we would memorize short conversations to give us a guide on how the words functioned. Our textbooks gave thorough explanations of the structures used as well, something Duolingo understandably lacks.
Was wondering the exact same thing... 涼しくin adverbial form, but さむい in adjective form?
the real question here would be: why are you adding a と there? are you trying to connect both clauses with the と particle? If that's the case, you are already doing that with the ren'youkei of the i-adjective 涼しい which transforms into 涼しく in order to be used as a conjunction. I made another comment in the same thread with more about this.