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Relation to Other Ways
The fifth way is grouped with the fourth way in demonstrating the perfection of God. This specifically covers more intimately the perfection of the operation of the first cause in that He orders all things intelligently.
Terms
The terms of this proof are much less than previous ones. We have already built a foundation in concepts such as limited regress, so this inevitably is going to assume some prior knowledge.
The only concepts that are that of "natural thing" and "finality."
Natural thing: Anything lacking intelligence.
Finality can be of two types. This is central in understanding the argument. Some have haphazardly stated this argument as teaching that God exists because of universal or intrinsic order. Rather, this argument takes its foundational premise from the fact of the finality of INDIVIDUAL AGENTS in their INDIVIDUAL ACTS.
Thus, we have a distinction between two types of finality. First, there is external finality. This would be asking "what is the purpose behind the existence of cancer" in relation to the whole of creation. This is a much more robust claim that would require many layers of argumentation. This is not the claim of St. Thomas here.