Členy jsou něco, co mi i po letech denního užití AJ dělá problém, tak jsem si vybral pár vět a prosím o vysvětlení:

• We see that as a first step. –proč ne “the”? Chápu, že někdy je to první jako obecně (when you choose a first car), ale to rozlišení mi prostě nejde.
• First, let's define the/an approach to classification of... – chci definovat jeden určitý přístup, ale zároveň o něm mluvím poprvé.
• Hence you should adjust your processes, tools, and metrics in a way that fits the/a project best. – řeším nastavení v projektech, nemluvím o žádném konkrétním projektu ale v tomto případě mi “the” sedí lépe, ačkoli je to první zmínka i něco co platí pro každý libovolný projekt.

Díky předem

1. We see that as a/the first step. Either version is possible, see the Ngram.

• A first step = the usual “first mention” indefinite article, without any further considerations in the speaker's mind.
• The first step = at the time of speaking, the speaker already envisages (AmE envisions) the prospect of further steps.

2. First, let's define the/an approach to … Either version is possible, see the Ngram.

• an approach = the usual “first mention” indefinite article, without any further considerations in the speaker's mind.
• the approach = at the time of speaking, the speaker already has in mind a definite approach, even if it's not yet stated. He's thinking of the approach that he will use, or the approach that's been successful before, or some other defined approach. In his head, the approach is defined.

3. Hence you should adjust your processes, tools, and metrics in a way that fits the/a project best. Either version is possible.

• … fits the project = it's a project which has already been identified or which we already have in mind. This is the more likely option, because if you're already talking about tools, processes and metrics, you've probably got a specific project in mind: the project can be felt to be defined as the project you need the tools for, even if we don't know what that project is.
• … fits a project = any hypothetical project, not a particular project that we already have in mind.

(“Fits the project best”: 140 real hits on Google. “Fits a project best”: 8 real hits on Google.)

Thanks Dan. As for the “the/a” project, well actually I’m working on a guideline that should be applicable to all (future) projects in my department, it is rather generic and hence I would incline to use the indefinite article. It should help managers to set up basic procedures that are common to all projects regardless of their size. But it just sounds off to me