Weakly uniform computable categoricity and function composition

We introduce the definition of a weakly $X$-uniformly $Y$-computably categorical structure, relativizing a definition of Kudinov (1996), and show that this relativized notion is closely related to the 2-dimensional complexity of the pairs of partial functions $(f,g)$ with the property that the composition $g\circ f$ simulates convergent ${\mathrm{\varnothing }}^{\mathrm{\prime }}$-oracle computations. We develop a general theory for computing via composition and use it to prove results about the surprisingly complex weakly uniform categoricity spectrum of the structure $(\omega ,S)$ where $S(n)=n+1$. Namely, that $(\omega ,S)$ is w.$X$.u.$Y$.c.c. for some $X\mathrm{̸}{\ge }_T{\mathrm{\varnothing }}^{\mathrm{\prime }}$ and $Y{<}_T{\mathrm{\varnothing }}^{\mathrm{\prime }}$, but $(\omega ,S)$ is not w.$X$.u.$Y$.c.c. for any $X{<}_T{\mathrm{\varnothing }}^{\mathrm{\prime }}$ and $Y\mathrm{̸}{\ge }_T{\mathrm{\varnothing }}^{\mathrm{\prime }}$. We also construct a computably categorical structure that is not w.${\mathrm{\varnothing }}^{\mathrm{\prime }}$.u.c.c., but is w.$X$.u.c.c. for a set $X$ that is ${\le }_T$-incomparable with ${\mathrm{\varnothing }}^{\mathrm{\prime }}$.