Products of posets in $\mathbf{\text{Pos}}$ as a special case of products of categories in $\mathbf{\text{Cat}}$? 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

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 $C \times D$.
