Re: the discussion on the word "obvious." 1. I agree that it has an important (for my working style, extremely important... the most common question I ask to people is "Is this trivial?" with the expectation of a yes/no answer and nothing more) role. 2. The best solution (when explaining to others) I've found for this is to use the word "formal" in place of obvious, combined with the right tone of voice. This also allows for the use of phrases like "there's a tricky formal argument for this".

There is the objection that "formal" and "obvious" have different meanings - but my opinion is that the majority of situations where they mean something different warrant more detailed explanations anyways. E.g. you could argue that the small object argument is formal but not obvious, but I would never use "this follows formally" or "this is obvious" when I mean "this follows from the small object argument."

The word I find harder to excise is "standard." (P.S. all of this is meant in the context of interpersonal explanations, where in my experience the word "obvious" is used differently than in a paper.)

like some others have shared here, i feel like the only times i've ever used the word "obvious" have been when i don't understand the situation, and i often end up being wrong. i do only use "straightforward" when i have actually worked out the proof and deemed it unnecessary to include so as not to clutter the exposition, and in that case i add some indication of what to do in the verification (for instance, "this is a straightforward verification by using the adjunctions" etc.)

There's the story of the professor giving a lecture, and a grad student asks: "On the other board there you say [...]. Is that obvious?" The professor starts to explain, but stops suddenly. They start pacing in front of the blackboard in silent contemplation. Time stretches on, and soon people in the audience start to trickle out, leaving the professor alone, pacing. In the next class, the professor arrives, a few moments late, announcing to the room, "It is obvious, and here's why..."

from the rest of this discussion, i think my personal opinion is that "obvious" shouldn't be used in exposition meant to be read by other people, as if it really is obvious one shouldn't need to say so. i like @dhy's ideas about the word "formal", especially e.g. if something follows formally from the small object argument, saying "this follows formally from the small object argument" rather than "this follows formally".

i'd want to change my usage of "straightforward" to be more of a descriptor of a proof to follow than an excuse to omit a proof, though i'm on the fence here.

the difficulty i have, not being an expert, is knowing when it is reasonable to omit some details and in what context. but i guess that comes with experience and asking other people!