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I am reading Awodey's Category Theory. In §2.5 he mentions that from the the product category, which is defined in $\mathbf{\text{Cat}}$, we can obtain a product of posets in $\mathbf{\text{Pos}}$ as well:

As special cases, we also get products of posets and of monoids as products of categories. (Check this: the projections and unique paired function are always monotone and so the product of posets, constructed in Cat, is also a product in $\mathbf{\text{Pos}}$, and similarly for $\mathbf{\text{Mon}}$.)

But I find this so puzzling!

Let $P$ and $Q$ be two posets viewed as categories, that is, $P,Q \in \mathbf{\text{Cat}}$. I know from the definition of the product category that $P \times Q$ is well-defined in $\mathbf{\text{Cat}}$. Suppose I have also shown that the projection and unique paired function are monotone. How to "transport" it to $\mathbf{\text{Pos}}$ now? I mean, how can I conclude that $P \times Q$, that lives in $\mathbf{\text{Cat}}$, lives in $\mathbf{\text{Pos}}$ as well?

All this "transport" confusion has lead me to the slightly general question:

Side Question:

If I have a poset $P$ (or a monoid $M$) in $\mathbf{\text{Cat}}$, how can I define formally its "counterpart" $P'$ in $\mathbf{\text{Pos}}$ (or $M'$ in $\mathbf{\text{Mon}}$)? I guess we should show they are isomorphic by defining a functor "from the former to the latter". But this seems so loose and confusing. To begin with, they aren't even objects of the same category, so how can I do such a thing?

It would be great if you could help me to understand this topic that is making me loose sleep!

Thanks

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Preorders are precisely those small categories in which every diagram commutes. If every diagram in $C$ resp. $D$ commutes, then this is clearly true for $C \times D$.

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