This $\ast$ operation can appear when you count the number of natural transformations between polynomial endofunctors on $C$. For example, if we abuse notation so that $f$ is both a univariate polynomial and its corresponding polynomial endofunctor, then $|Nat(f,1_C)|=f^\ast(1)$.

I came across this recently when writing code to memoize polymorphic functions.

This $\ast$ operation can appear when you count the number of natural transformations between polynomial endofunctors on $Set$. For example, if we abuse notation so that $f$ is both a univariate polynomial and its corresponding polynomial endofunctor, then $|Nat(f,1_{Set})|=f^\ast(1)$.

I came across this recently when writing code to memoize polymorphic functions.

This $\ast$ operation can appear when you count the size of sets of natural transformations between polynomial endofunctors on $C$. For example, if we abuse notation so that $f$ is both a univariate polynomial and its corresponding polynomial endofunctor, then $|Nat(f,1_C)|=f^\ast(1)$.

I came across this recently when writing code to memoize polymorphic functions.

This $\ast$ operation can appear when you count the number of natural transformations between polynomial endofunctors on $C$. For example, if we abuse notation so that $f$ is both a univariate polynomial and its corresponding polynomial endofunctor, then $|Nat(f,1_C)|=f^\ast(1)$.

I came across this recently when writing code to memoize polymorphic functions.