Let $C$ be the category of sets $X$ together with a map $\phi:X^{\mathbb{N}} \to X$, where $\mathbb{N}$ is the set of positive integers (which does not include zero).
Some properties we might consider include:
- The forgetful functor $C \to \mathbf{Set}$ has a left adjoint.
- The forgetful functor is monadic.
- The category $C$ is cocomplete.
- The category $C$ is regular (but perhaps, the image of a morphism might differ from the usual one).
- The category $C$ is Barr-exact (but perhaps, the quotient by an equivalence relation might differ from the usual one).
- The image of any morphism in $C$ is a subalgebra (which implies regularity).
- For any object $(X, \phi)$ of $C$ and any equivalence relation $E$ on $X$ such that for any two sequences $(x_n)_{n \ge 1}$ and $(y_n)_{n \ge 1}$ with $(x_n, y_n) \in E \forall n \in \mathbb{N}$, one has $(\phi((x_n)_{n \ge 1}), \phi((y_n)_{n \ge 1})) \in E$, the quotient $X/E$ has a natural induced structure as an object of $C$ (which implies Barr-exactness).
Question:
Is each of the properties of the category $C$ above equivalent to the axiom of countable choice?