'Contravariance has more mathematical structure in it whereas covariance is more geometrical and easy to understand.' Is it a serious claim? In Shastri's Basic Algebraic Topology Remark 1.8.8, the author wrotes that

The difference between covariance and contravariance is simply in the fact that covariance preserves the direction of the arrow whereas contravariance reverses it. However, in practice, it turns out that contravariance has more mathematical structure in it whereas covariance is more geometrical and easy to understand.

I am really curious as to why the author thinks that way. Unfortunately, he doesn't provide further justification of that. My question is:


*

*Is this a serious claim? (i.e. Is there a sense in which it is true?)

*If the answer to 1 is 'yes', what's the underlying reason for this phenomenon? (maybe there is a categorical justification?)


Though I am just a beginner in algebraic topology, answers from all levels are welcomed.

As Connor Malin comments below, every contravariant functor is a covariant functor from $\mathsf{C^{op}}$ to $\mathsf D$, so the two concepts are duals of each other, which further mystifies the claim.
 A: Is it a serious claim? I claim this doesn't have a serious answer: It is opinion-based.
Remark 1.1.8 follows the definition of covariant and contravariant functors, thus it is certainly a statement about functors used in practice. There are other contexts where the words "covariant" and "contravariant" are used, for example in the context of (multi-)linear algebra; see Lutz Lehmann's comment and this question. I guess that most people (including physicists and engineers) who have heard these word associate it with tensors. But this context is irrelevant here.
Have a look at the contents of the book. It has 13 chapters, but one cannot say that contravariant functors (essentially cohomology functors) are overrepresented and provide more mathematical structure than covariant functors.
There are some nice things which you can do with cohomology, for example you can introduce the cup product and get the cohomology ring or define cohomology operations. But all that can be dualized, for example you can give homology a coring-structure. Many people have never heard about this concept, thus it may appear somewhat strange.
Contravariant functors are often representable ("Brown's representability theorem") which led to the fruitful concept of a spectrum. Another example is $K$-theory which arises via classifying vector bundles on spaces $X$; this gives a very important (generalized) cohomology theory. The construction is geometrical and easy to understand although it is contravariant.
This shows that many important concepts of algebraic topology are based (in a natural way) on contravariance. However, other things like homotopy groups are covariant.
I think we need on equal terms both co- and contravariance. The whole complex of duality, for example duality in manifolds, relies on both types.
