A forgetful functor ignores structure on a set (under an embedding interpretation) prior to its mapping to another set. So, if we have a set with signature: (ℝ,+,α,β). Some forgetful functor may map that to just (ℝ,+). This seems understandable enough. But, if one takes something fanciful, such as one of the various lemmas or theorems (c.f. Yoneda, Cayley's representation) that embed arbitrary structures into collections of subsets, or the equivalent, it brings one to another view of this functor. Say, we have an object which has been embedded into a set so as to look like ordered pairs: {{x(adsbygoogle = window.adsbygoogle || []).push({}); _{1},y_{1}},{x_{2},y_{2}},...,{x_{n},y_{n}}}. Is it admissible to see this forgetful functor as choosing to ignore one of the terms of the ordered pair as it maps each element of the collection to the target?

This seems important to me, as many works on category theory first introduces Set, where it demands that each morphism (in that particular category only) is nothing short of atotalfunction. Though, most don't say total function explicitly: they'll generally say that it must map each element of the domain to one or more elements in the codomain. This is non-trivial to me; because, functors are "just" morphisms as well. And, according to the theorems I mentioned above, nearly all categories can be represented in Set. So, I am trying to find how to consolidate my understanding. The functor seems to "break" this rule by arbitrarily choosing some elements of the embedded set.

Lastly, there is the concept of a section and a retraction from category theory. I am wondering what the relation is to what I mention above. That is, they come up in something called the "choice" and "determination" problems. Isn't that similar to the "choice" being made by the forgetful functor, in the context of the embedding example, when only part of the structure (embedded signature) is mapped? Isn't it, in reality, a partial function? Are sections/retractions related to partial functions?

Thanks in advance for any help. And, also for your patience. My terminology is not up to par, I realize.

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# The partial function, the section/retraction, and the functor

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