Six object classes | products | co-products in search of a category

Question

^{(Apologies to Pirandello for the title.)}

I have a couple of quite analogous situations, each featuring candidates for the objects, products, and coproducts of a possible category; missing still are the morphisms¹ (including, of course, the identities) that would yield the corresponding bona fide categories.

I will refer to these hypothetical categories as $\mathbf{C}_1$ and $\mathbf{C}_2$.

1. $\mathrm{ob}(\mathbf{C}_1) = \mathbb{N_0}$, the set of non-negative integers, and the product and coproduct are the usual multiplication and addition in $\mathbb{N}_0$.

2. $\mathrm{ob}(\mathbf{C}_2) = \mathrm{ob}(\mathbf{Set})$, and the product and coproduct are the usual set-theoretic intersection and (not-necessarily-disjoint) union.

Is it possible to define morphisms for either of these cases that would render it a category? Also, if there's a standard name (as a category) for any of these cases please let me know.

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Regarding the first case, I find it suggestive that $\#$ (cardinality²) maps $\mathrm{ob}(\mathbf{FinSet}) \to \mathbb{N}_0$ in such a way that $$\#(A\;\Pi\;B) = \#A \times \#B,$$ and $$\#(A \amalg B) = \#A + \#B.$$ Therefore, if there is indeed a category $\mathbf{C}_1$, it looks like $\#$ would be a good candidate for a functor $\mathbf{FinSet}\to \mathbf{C}_1$. In fact, a suitable extension of $\#$ to the morphisms of $\mathbf{FinSet}$ would provide a pretty good idea of what the morphisms of $\mathbf{C}_1$ would have to look like.

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Addendum (in response to Qiaochu Yuan's answer). For example, I can see how the object called³ $\mathbf{18} = \{0,\dots, 17\} \in \mathrm{ob}(\mathbf{C}_1)$ could serve as the "common domain" of two surjections (i.e. "candidate projections") $\mathbf{18} \rightarrowtail \mathbf{3}$ and $\mathbf{18} \rightarrowtail \mathbf{6}$ (where $\mathbf{3} = \{0, \dots, 2\}, \mathbf{6} = \{0, \dots, 5\} \in \mathrm{ob}(\mathbf{C}_1)$), but it's not clear to me how one specifies these surjections in detail so that they constitute a categorical product $\mathbf{3} \; \Pi \; \mathbf{6}$.

(BTW, I think part of my difficulty here is that it's hard for me to know when the phrase "up to isomorphism" is being left out, or what exactly this tacit isomorphism is.)

Addendum 2 OK, I see now: one way to define $\mathbf{18} \rightarrowtail \mathbf{3}$ and $\mathbf{18} \rightarrowtail \mathbf{6}$, respectively, could be $n \mapsto \lfloor n/6\rfloor$ and $n \mapsto n\;\mathrm{mod}\;6$. I had not expected an asymmetric definition, but I guess there's nothing wrong with it.

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^{1I realize that, when it comes to categories, having the objects, products, and coproducts, but not the morphisms, is a bit like having a table set with plates, silverware, napkins, etc., but no food: it may all be suggestive enough to whet one's appetite, but it's still a very long way from the real thing.}

^{2Originally I had used the notation $\mathrm{card}(\dots)$ for cardinality, instead of $\#$, but found that the font was too similar to the regular font, and it would get lost within the surrounding text.}

^{3Originally I had defined $\mathbf{18}$ as $\{1, \dots, 18\}$, but the current definition works better with the projections givein in Addendum 2.}

Answer 1

1. Take the category of finite sets and functions. This category is commonly called $\text{FinSet}$. (Strictly speaking, to satisfy your request I should take a skeleton of this category.)

2. Take the category of sets and inclusions (actually a large poset). I don't know if this category has a name; arguably it's usually not the category you want to work in, and you instead want to work in the category of sets and functions, which is commonly called $\text{Set}$. As far as I'm concerned, there's no such thing as the intersection or union of sets.