This is sort of true, but there are isomorphisms which aren't "good enough" in some sense. E.g. if you're familiar with linear algebra, there's a thing called a

This is sort of true, but there are isomorphisms which aren't "good enough" in some sense. E.g. if you're familiar with linear algebra, there's a thing called a dual space, which (for finite dimensions) is isomorphic to your space, but when you change coordinates, dual spaces change their coordinates backwards. Or for complex numbers, when you extend R by adding `i` (sometimes written C=R[i]), if you look carefully, you'll see that you could have instead added an element α = -i, and you'll notice that R[α] is isomorphic to R[i] in a way where the isomorphism maps R to itself; in a sense you can't tell whether you added i or -i, and this degree of freedom is important when studying field extensions. The closest thing I'm aware of to "pretty much equal" is "unique up to unique isomorphism", so they may not be equal, but they're isomorphic, and there's no flexibility to make any choices of which isomorphism to use. But "isomorphism" also implies you have a particular context/structure in mind that you want to preserve, e.g. a set isomorphism (a bijection) may not be a linear isomorphism (~an invertible matrix). In practice, you may be working in multiple contexts at once, so you invent Functors which map one type of morphism to another, and now you care about Categories.