I understand why terminal objects are "nullary products" in any category, and I understand why singleton sets are terminal objects in the category of sets. So I understand why singleton sets are nullary products in the category of sets.
Similarly, it seems like any object in any category equipped with the identity morphism as the "projection morphism" is a "unary product" (cf. the definition of product of arbitrary families). In particular, it seems that any set is a unary product in the category of sets.
Because singleton sets are sets, the second argument implies that singleton sets are unary products, while the first argument implies that singleton sets are nullary products. So is it really correct to say that singleton sets are simultaneously both nullary products and unary products?
Or is it more correct to say that:
- a singleton set without any associated morphisms is a nullary product (but not a unary product), whereas
- a singleton set together with its identity morphism is a unary product (but not a nullary product)?
Comment: I suppose this is no more confusing than how, given an arbitrary set $X$, the set $X \times \{ \ast \}$ is both a unary product for $X$ using $\pi_X$ only, but also a binary product for $X$ and $\{ \ast \}$ using both $\pi_X$ and $\pi_{\{\ast\}}$. I guess part of me wants to somehow give objects "an intrinsic -arity", even though categorical products are defined in terms of an object together with associated projection morphisms, not just as an object alone. There is no rule that projection morphisms in the definition of a product can't be constant functions, so there is nothing that excludes projections onto singletons.
Although if what I'm saying is true, then it seems to contradict the premise of this other question, because I am essentially claiming that singleton sets could be used to create product structures of arbitrary "-arity", at least as long as one allows products with projection morphisms that are constant functions.
One and the same object may participate in multiple universal constructs. For example, for the two-element set $2=\{a,b\}$, we have that $2\times X$ equipped with projection morphisms $2\leftarrow 2\times X\to X$ is a product. But equipped with the morphisms $X\to 2\times X$ given by $x\mapsto(a,x)$ and $x\mapsto(b,x)$ it is also the coproduct of $X$ with itself.
Nullary products and coproducts are slightly exceptional in that their universal structures are empy, but they can have other universal structures as you observe. In any case, it's more correct to say that the singleton equipped with the identity morphism is a unary product.
Your observation that you can create products of arbitrary arity with projections onto unary products is correct; it does not contradict the linked question because you can't create products of arbitrary collections of objects: you only increase arity by including more and more projections onto nullary products.
By the way, a unary product of an object $X$ is an object $Y$ equipped with an isomorphism $Y\to X$.