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An inner join is a Cartesian product with conditionals added.

There's a big performance difference between creating the Cartesian product and then filtering for the conditional, and creating the conditionals directly. An inner join with equi join conditions creates the conditions directly; any non equi join conditions actually have to be evaluated

That's true, but that is an implementation detail. In abstract terms, you can think of an inner join like that.

Also, while what you say is true in general for modern DB's, there are some implementations like old Oracle versions where the only way to create the effect of an inner join was in terms of a Cartesian product.

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