TITLE: Relationship between $ \textbf{Cat}$ and $ \textbf{Ord}\ $?
QUESTION [0 upvotes]: I am trying to understand the inter-relationships between the following four structures/categories.  (This question is similar to problem 8 in Awodey's book, page 26.)
(1) Individual categories (which I designate as $ \textbf{A}$, $ \textbf{B}$ , $ \textbf{C}$ , etc.).
(2) Individual categories of preorders induced by those (single) categories (which I designate as $ \textbf{P(A)}$, $ \textbf{P(B)}$, $ \textbf{P(C)}$, etc.).  These latter categories have the same objects as the the first categories, and morphisms $ A \leq B $ iff there is an arrow $ A \to B $  in the first initial category.
(3) The category of all categories, $ \textbf{Cat}$, and 
(4) The category of all pre-ordered sets, $ \textbf{Ord}$.  This cat has pre-ordered sets (see #2 above) as objects and order-preserving functions as morphisms.
The latter two categories are simply the two former categories formed by 'aggregation" of the first two categories.
Now, the relationship between (1) and (2) is rather straightforward and just as described above, with a functor we'll call $ \textbf{P}$ making the obvious transformation.  It appears to me that this is also a surjection.
What I am wondering is how to form a functor induced by P from the 
category of all categories $ \textbf{Cat}$ to the category of all pre-ordered sets, $ \textbf{Ord}$ .  The objects are easy, but how would I map the general morphisms/functors in $ \textbf{Cat}$ to the order-preserving functions needed in $ \textbf{Ord}\ $ ?
P.S. I know that as single categories both $ \textbf{Cat}$ and $ \textbf{Ord}$ fit into my first collection of categories.  For now, I'd like to ignore that complication if possible.

REPLY [3 votes]: The relationship between $A$ and $P(A)$ is different from what you claim: $P$ loses lots of information (in particular it's far from being bijective, in any sense). However, for each $A$ there is an obvious functor  $F_A: A\rightarrow P(A)$ which is the identity on objects, and moreover there is a simple description of a corresponding functor from Cat to the category of preorders, which I will call Pre (Ord, I think, refers to the category of ordinals - at least that's how I've seen it used in the past). 
Namely, given a functor $\mu: A\rightarrow B$, we want to get an induced order-preserving map $P(\mu): P(A)\rightarrow P(B)$. But this $P(\mu)$ can just be the map $\mu$ itself (restricted to objects)! Suppose $a_1\le_{P(A)}a_2$. Then there is a morphism $m:a_1\rightarrow a_2$ in $A$. Let $\mu(m):\mu(a_1)\rightarrow\mu(a_2)$ be the corresponding morphism in $B$. Then $\mu(a_1)\le_{P(A)}\mu(a_2)$, since $\mu(m)$ exists. So $\mu:P(A)\rightarrow P(B)$ is an order-preserving map. (I'm abusing terminology a bit here, but it should be clear what's going on.)
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