ct.category theory - Can the inner structure of an object be systematically deduced from its position in the category?

Background

Even for the novice it seems comprehensible that the "inner structure" of an object is determined (up to isomorphism) by its "position" in a category, defined by the morphisms.

What is not so obvious is how the inner structure of an object can be recovered from its position.

Question

What has to be given (and how) to reconstruct the inner structure of an object (up to isomorphism)? What conditions must the category fulfill? And how could the reconstruction be systematically achieved?

Examples

The following examples are simple in the sense that the categories are especially tailored: the objects are finite, there are no isomorphisms (except the identities) and all hom-sets contain at most one element. The question is, whether similar reconstructions can succeed in the general case, too.

(1)

Consider the category of "unlabeled" finite sets (vulgo natural numbers with a sequence of Hilbert-strokes as "inner structure") with the $leq$ relations as morphisms ($leq$ means "injectively embeddable", the number of possible embeddings ignored). Now consider only identities and prime morphisms (see my definition), corresponding to the relation $x = y + 1$ ("x is reachable from y by adding one stroke").

It is easy to "see" the inner structure from the position of an object with respect to the initial object (the empty set).

(2)

Consider the category of finite undirected unlabeled graphs without isolated vertices. The morphisms correspond to the relation "is edge-wise incidence-preserving embeddable" (ignoring again the number of possible embeddings). Consider again only identities and prime morphisms, corresponding to the relation "x is reachable from y by adding one edge".

See a fragment of this category here (identities and arrow heads not displayed).

Conjecture: Each object in this category is uniquely determined by the tuple $(n,k,l)$ with $n$ its distance from the initial object (the empty graph), $k$ its number of in-going morphisms and $l$ its number of out-going morphisms.

Claim: The inner structure of each object up to distance 3 from the initial object can be systematically deduced from its position, considering (only?) its tuple $(n,k,l)$.

Observation: This category reflects something like the "generating lattice" of the graphs (each non-identity morphism corresponds to adding one edge). Which other categories can be interpreted in this way?