On elements in category theory

Today I stumbled upon a quote by Lawvere:

There has been for a long time the persistent myth that objects in a category are “opaque”, that there are only “indirect” ways of “getting inside” them, that for example the objects of a category of sets are “sets without elements”, and so on. The myth seems to be associated with an inherited belief that the only “direct” way to deal with whole/part relations is to write an unexplained epsilon or horseshoe symbol between A and B and to say that A is then “inside” B, even though in any model of such a discourse A and B are distinct elements on an equal footing. In fact, the theory of categories is the most advanced and refined instrument for getting inside objects, because it does provide explanations (existence of factorizations of inclusion maps) and also makes the sort of distinctions that Volterra and others had noted were necessary for the elements of a space (because the elements are morphisms whose domains are various figure-types that are also objects of the category)

This is a companion discussion topic for the original entry at https://matteocapucci.wordpress.com/2023/08/21/on-elements-in-category-theory/
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@Eigil had a great use of generalized elements in a localcharts post from a while back: Equivalence relations via idempotent endomorphisms.

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