Your $\mathfrak g\circ\mathfrak f$ looks as if it will be an ultrafilter if at least one of the factors $\mathfrak f$ and $\mathfrak g$ is principal, or more generally if, for some cardinal $\kappa$, one of the factors contains a set of cardinality $\kappa$ while the other is closed under intersections of $\kappa$ sets at a time. But it won't be an ultrafilter in general, not even if $\mathfrak X=\mathfrak Y=\mathfrak Z=$ the additive monoid of natural numbers (viewed as a one-object category).

For this and (lots) more information about adding ultrafilters on $\mathbb N$ (and generalizations), the standard reference is the book "Algebra in the Stone-Cech compactification" by Neil Hindman and Dona Strauss. In a combination of the book's notation and yours, both $\mathfrak f+\mathfrak g$ and $\mathfrak g+\mathfrak f$ are ultrafilters extending $\mathfrak g\circ\mathfrak f$, and they are not always the same ultrafilter by Theorem 4.27.

Your $\mathfrak g\circ\mathfrak f$ looks as if it will be an ultrafilter if at least one of the factors $\mathfrak f$ and $\mathfrak g$ is principal, or more generally if, for some cardinal $\kappa$, one of the factors contains a set of cardinality $\kappa$ while the other is closed under intersections of $\kappa$ sets at a time. But it won't be an ultrafilter in general, not even if $\mathfrak X=\mathfrak Y=\mathfrak Z=$ the additive monoid of natural numbers (viewed as a one-object category).

For this and (lots) more information about adding ultrafilters on $\mathbb N$ (and generalizations), the standard reference is the book "Algebra in the Stone-Čech compactification" by Neil Hindman and Dona Strauss. In a combination of the book's notation and yours, both $\mathfrak f+\mathfrak g$ and $\mathfrak g+\mathfrak f$ are ultrafilters extending $\mathfrak g\circ\mathfrak f$, and they are not always the same ultrafilter by Theorem 4.27.