Motivation
The structuralist point of view on mathematical objects has two aspects:
On the one side, a mathematical object is seen as a concrete structure of abstract dots, e.g. a graph.
On the other side, a mathematical object is seen as an abstract dot in a concrete structure (of abstract dots), e.g. a category.
Most mathematical objects fit easily into both of these pictures: a graph, group, ring, field, etc. can be seen as a concrete structure of abstract dots, and as an abstract dot in a concrete structure, e.g. a category.
Even a single natural (= finite cardinal or ordinal) number fits into both pictures: as a specific dot in the infinite structure of natural numbers $\bullet\rightarrow\bullet\rightarrow\bullet\rightarrow\dots$, or as a bag of dots $\lbrace\bullet\bullet\dots\bullet \rbrace$, resp. a finite initial segment $\bullet\rightarrow\bullet\rightarrow\dots\rightarrow\bullet$ of the natural numbers.
The same goes e.g. for hereditary finite sets.
The natural numbers $\mathbb{N}$ as a whole do fit also: as an abstract dot in some category, and as the concrete structure $\bullet\rightarrow\bullet\rightarrow\bullet\rightarrow\dots$
Question
For some objects I find it harder to fit them into both pictures. While it's easy to imagine an abstract ring that is isomorphic to a polynomial ring - with its dots representing "polynomials" - I have no idea what a polynomial is as a concrete structure (of abstract dots) in the sense above - or at least as a equivalence class of such concrete structures, equivalent with respect to the relation of "representing the same polynomial".
So:
Is there something like a polynomial as a concrete structure (of abstract dots)?
Or why is this question misled?