It is well known (if you're a topos-theorist, you will call it the definition), that the natural numbers $\mathbb{N}$ together with the zero constant $0$ and the successor function $1\xrightarrow{\ 0\ }\mathbb{N}\xrightarrow{+1}\mathbb{N}$ are initial in the category of such diagrams of sets $1\longrightarrow X\longrightarrow X$ (this is known as the principle of inductive definition).
Suppose we work in some class theory like NBG.
Is it true that the natural numbers are still initial in the (meta-)category of such diagrams of classes? Or in set-theoretical terms, does inductive definition work for classes?
Originally, I was thinking about the following: Suppose we replace the classical Axiom of Infinity by the
Axiom of an NNO: There exists a Natural Numbers Object in $\mathbf{Set}$.
Can I show that finite cardinals form a set by recursively defining a function $\mathbb{N}\xrightarrow{\ \ f\ \ }(\text{Class of all Sets})$ by $f(0)=\emptyset$ and $f(n+1)=f(n)\cup\{f(n)\}$ and then applying the Axiom of replacement?
Start by assuming $\omega$ is the smallest set containing $\emptyset$ and closed under the operation $x \mapsto x \cup \{ x \}$. (Such a thing is guaranteed to exist in ZFC and NBG.) We want to show that $\omega$ is an NNO in the category of classes.
Let $X$ be a class, let $x_0 \in X$, and let $F : X \to X$ be a class-function. By the axiom of class comprehension, we may form the class $A$ of all partial functions $f : \omega \to X$ such that $f (\emptyset) = x_0$ and $f (x \cup \{ x \}) = F (f (x))$. (Of course, by ‘partial function’ I mean a set of pairs satisfying the obvious conditions.) It is non-empty: the partial function $\{ (\emptyset, x_0) \}$ is a member of $A$. Moreover, by ordinary induction, we can show that, for every $n \in \omega$, there is $f \in A$ with $(n, x_n) \in f$ for some $x_n \in X$; moreover, since $F$ is functional, for every $n \in \omega$, there is a unique $x_n \in X$ such that $(n, x_n) \in f$ for all $f \in A$ such that $(n, x) \in f$ for some $x \in X$. This defines a class-function $\omega \to X$ (by class comprehension), so we are done.
It appears to me that the above goes through for ZFC, provided we interpret "class" and "class comprehension" correctly.