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The simplest case is that for a fixed (small) category $C$, there is a (multi-sorted) first-order theory whose models are functors $C\to \rm Set$: it has one sort for every object of $C$, and one func …

I could be wrong (it's hard to be sure since the phrase "preserves inverses" is a bit vague), but I think that's why they said they were defining only a strict action. A weak action would only preser …

Yes. The map $G_{A'} \to F_f \circ G_A$ is called the mate of $T_f : F_A \to F_{A'} \circ F_f$. It's the composite $$ G_{A'} \to G_{A'} F_A G_A \to G_{A'} F_{A'} F_f G_A \to F_f G_A$$ of $T_f$ wit …

On the other hand to Tom's answer, the distinction between strong and weak equivalences disappears if you use anafunctors instead of functors, which is arguably the "right" way to do category theory i …