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The relevant statement in Adamek and Rosicky is: a functor $G: C \to D$ between locally presentable categories is a right adjoint if and only if it preserves limits and is accessible (meaning it preserves $\lambda$-directed colimits for some regular cardinal $\lambda$.
I can make an adjustment to remove doubts.
Okay, I removed the ambiguity in language, to make clear we were indeed talking about logical necessity. Please have a look. (And thanks for pointing out how it read.)
I usually write “precisely if” instead of “if and only if”, as it seems to be more decent prose to me. I wasn’t aware that it comes across as ambiguous.
Normally I don’t think it would be taken as ambiguous. But questions were raised about it after Hurkyl was trying to put it all together with other stuff on that page. So I was trying to remove every last vestige of doubt by using language whose meaning cannot be misread.
I would probably consider “if and only if” to be more precise than “precisely if”, and it doesn’t seem at all indecent to me. (-:
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