I am wondering the meaning for a functor to have a left adjoint or a right adjoint in practice.
Stardard results including Freyd's adjoint functor theorem (preserving colimits or limits) are not really helpful for my understanding. Also, if we have two adjoint functors forming an equivalence, the two categories are completely symmetric. But for adjunction, the unit and counit maps only have natural transformation one way. I want to understand more about this subtlety here.
For example, when I read this wiki page, the inclusion functor $$\iota_{\geq n}: \mathscr{D}_{\geq n}\to \mathscr{D}$$ admits a right adjoint, but the other inclusion functor $$\iota_{\leq n}: \mathscr{D}_{\leq n}\to \mathscr{D}$$ admits a left adjoint. I cannot understand why it is not the reverse way.
A reformulation of this question could be as follows. Given two functors are adjoint, how can one tell which is the left adjoint and which is the right one?
An example that is still unclear to me is the Kerodon page stating that the nerve functor is right adjoint to the homotopy functor. Why it is not left adjoint?
Since this is a general question, the answer does not have to be in the context of $\infty$-categories. For example, the covariant hom functor $\text{Hom}(A,\_)$ has a left adjoint, which is the tensor product, while the contravariant hom functor $\text{Hom}(\_,A): \mathscr{C^{op}}\to \mathscr{D}$ preserves limits, therefore under some conditions has left adjoint which turns out to be the opposite of itself (c.f. here).
I heard from some seniors saying that in practice existence of right adjoint is more common than left adjoint for a functor, so colimits are often preferred. I would be grateful if someone can justify this too.
