I think of the core phenomenon as a special case of a basic fact about finite transformation monoids. Suppose we have a concrete category of finite structures such that bijective endomorphisms are isomorphisms and the image of each endomorphism is a substructure. Then for each object $X$ there is a subject $Y$ which is a retract of $X$, any endomorphism of $Y$ is an isomorphism and $Y$ is unique up to isomorphism. Moreover, $Y$ is of minimal cardinality among images of a morphism from $X$ to itself.
Why? Let $M$ be the endomorphism monoid of $X$. Define the rank of an element of $M$ to be the cardinality of its image. Let $m\in M$ have minimal rank $r$. Then all powers of $m$ have the same rank and hence there is an idempotent $e$ achieving the rank $r$. Let $Y=eX$. Then $Y$ is a retract of $X$. If $f$ is an endomorphism of $Y$, then $ife$ (where $i$ is the inclusion) is an endomorphism of $X$ and so by minimality of the rank of $e$, we have that $|eX|=|Y|\leq |fY|=|feX|\leq |eX|=|Y|$ and thus $f$ is an automorphism of $Y$ by our hypotheses.
Suppose $Z$ is another retract of $X$, say by an idempotent $e'$, such that every endomorphism of $Z$ is an isomorphism. Then $e'|_Y\colon Y\to Z$ and $e|_Z\colon Z\to Y$ are morphisms by restriction and hence $e|_Ze'|_Y$ is an automorphism of $Y$ and $e'|_Ye|_Z$ is an automorphism of $Z$. It easily follows that $e'_Y$ is an isomorphism from $Y$ to $Z$.