See this reference, under the question 'Whether it can be demonstrated that God exists'.

It would be useful in future if you provided a link or reference for further context for weighty questions such as these.

I note that this is mentioned as the second objection to the proposition. The answer is given below as:

Reply Obj. 2: When the existence of a cause is demonstrated from an effect, this effect takes the place of the definition of the cause in proof of the cause’s existence. This is especially the case in regard to God, because, in order to prove the existence of anything, it is necessary to accept as a middle term the meaning of the word, and not its essence, for the question of its essence follows on the question of its existence. Now the names given to God are derived from His effects; consequently, in demonstrating the existence of God from His effects, we may take for the middle term the meaning of the word God.

Objection 2 reads, “The essence is the middle term of demonstration,” not, “The essence of a demonstration is the middle term.” For example:

• All dogs are animals
• All animals are warm-blooded
• Therefore, All dogs are warm-blooded

The essence (“what it is”) of ‘animal’ is the middle term of the syllogism.

Aquinas responds to the objection by noting that one never appeals to an essence when proving the existence of a cause from an effect. This is because an essence presupposes existence and therefore when proving existence one cannot make use of essence as the middle term, but must instead use, “the meaning of the word.” This is a rather important aspect of Aquinas’ proofs for God’s existence, often overlooked.

Aquinas responds to the objection by noting — Leontiskos

My gosh there's a lot in this. Please consider laying it out in a bit more detail? E.g., "animal," in your syllogism is the essence of both dog and warm-blooded, the latter two being existences? And thus this sample syllogism not an example of an argument/proof by cause-and-effect?

That syllogism represents the standard sort of case that the objector is appealing to. The essence of 'animal' is operative in both premises (and is the middle term). Dogs, animals, and warm-blooded things all exist, and because of this Aquinas would say that such premises based on an essence are legitimate moves (i.e. Aquinas is not a Possibilist, and this school will not fully emerge until Scotus). And yes, the conclusion has to do with a property or characteristic of dogs, not their existence.

All dogs are animals
All animals are warm-blooded
Therefore, All dogs are warm-blooded — Leontiskos

I presume that you're using that rather old-fashioned taxonomy that doesn't count fish and reptiles as 'animals'? (This was quite common when I was growing up but nowadays 'animals' seem to cover everthing other than insects and possibly some inverterbrates.)

'animals' seem to cover everthing other than insects and possibly some inverterbrates — Wayfarer

Quick note: invertebrates (including insects) are animals according to standard use. That's the use I'm most familiar with, though of course I'm aware of the fading usage you mention (mammals).

That "Reply obj. 2" is some reply! Near as I can tell, it's petitio principii. Which, if the thing being begged is also an axiom, is not a fallacy. I find the problem at deducing causes from effects, and separately, the causes deduced from the effects.

- Yes, "mammal" would probably be the more commonly accepted term.