Learn English – “While you’re *at* it” – Why *at*

Strictly speaking, the sentence is incorrect because listening of playlists is ungrammatical (unless the playlists are doing the listening). Since categorization takes of and editing takes of but listening takes to, you have to write this:

Some websites allow categorization of, editing of, and listening to playlists online.

This is grammatically correct but it sounds very clumsy. If each noun took the same preposition, you could use the same preposition for all three:

Some websites allow categorization, editing, and playing of playlists online.

This is grammatically correct but it sounds even clumsier because of the repetition of play. People would rather make a subtle grammatical error than write a sentence that sounds this clumsy.

The fact that the first two of the nouns take of probably led people to ignore the incorrect listen of for almost ten years now.

Another “fudge” solution is to choose the preposition to agree with only the nearest noun even if it disagrees with all the others, known as “proximate agreement”:

Some websites allow categorization, editing, and listening to playlists online.

There is, however, a better way:

Some websites allow users to categorize, edit, and listen to playlists online.

This is clearer because the users are mentioned explicitly, and the nominalized verbs are replaced with plain old infinitive verbs. The preposition to only agrees with listen, but that's OK: categorize and edit are transitive verbs, which take an object without any preposition at all. So, to connects only with listen and there is no disagreement with categorize and edit. So, this version has perfect grammar as well as greater clarity.

By the way, many gerunds do take of. For example: editing of playlists, feeding of animals, planting of gardens, singing of songs, etc. Also, gerunds normally function as nouns. In the original sentence, editing and listening are objects of allow, just like categorization.

Unless I'm misunderstanding you, we don't use "to" the way you say... but I'm basing this on American English. Other varieties of English may use the prepositions differently.

To my understanding, this sentence means that cook something under the right temperature. Or that you need to know the right temperature of what you are cooking and cook it under that temperature.

I'll try to explain the usage of "cook to" in a way that should help

"Cook to" is usually seen in recipes when you're being told to cook something until it reaches a specific temperature or to bring the food's temperature up to a particular point:

Rack of Lamb: Rack of lamb should be cooked at 325 degrees. For rare; cook approximately 1 hour or to an internal temperature of 130 degrees. For medium rare; cook to an internal temperature of 140 degrees. [emphasis added]

In this example, the internal temperature is what tells you that the lamb has reached the preferred level of doneness and even the amount of time it cooks and the temperature of the oven can't guarantee that it will have reached that internal temperature, so it is necessary to use a thermometer.

Here's a great image of a rack of lamb with a meat thermometer in it... as you can see in the image, the internal temperature is just above 130 F.

Image from here.

This is likely using definition 10 of "to" here:

10 a) as far as a particular point or limit:

• Temperatures dropped to 25 degrees below zero.

As someone who cooks and reads a lot of recipes, this is the only definition of "to" that works here. Even metaphoric phrases make the same definition of "to":

The steak was cooked to death. (the steak was really overcooked)
The steak was cooked to perfection. (the steak was cooked perfectly)

In fact, the other way that this is phrased is

Cook until it reaches an internal temperature of 140 degrees.

When you say something should be cooked "under" the right temperature, I think you are correct to believe it should be "at" or "in". Let's go back to the example I posted above but look at a different sentence this time:

Rack of Lamb: Rack of lamb should be cooked at 325 degrees. For rare; cook approximately 1 hour or to an internal temperature of 130 degrees. For medium rare; cook to an internal temperature of 140 degrees. [emphasis added]

So, in this sentence, when it's being explained what temperature you need to set your oven for, we use "at".

This happens to also be definition 10 for the preposition "at":

10 used to show a price, rate, level, age, speed etc:

• The Renault was traveling at about 50 mph.

I think of it as "the oven is set at 325 degrees".

With slight rephrasing, we also often use "in":

Rack of lamb should be cooked in a 325 degree oven.

And for this one, you can think of the fact that you're putting the food inside an oven that is 325 degrees.

As to "under", there's only one example I can think of where "under" would be appropriate instead of "at" or "in" and that's in the case of using a broiler.

A broiler is a special heating element in an oven that is designed to cook food from above. Placement of heating units in ovens varies by oven but broilers are always above the food when actively broiling.

In the United States, when the heat source for grilling comes from above, grilling is termed broiling.

Because of this, we regularly say "under the broiler".

3. Spray a baking sheet with cooking spray (or coat lightly with olive oil) to prevent sticking. Lay the chicken breasts side by side not allowing them to touch each other. Cook under the broiler for about 5 minutes on each side, or until slightly charred and cooked through. Remove the chicken from the oven, baste with barbecue sauce, and broil for another minute. Remove from the oven and serve.