Is it possible to have a preorder with different kinds of morphisms? I am a non mathematician, quite new to category theory and would have the following question:
Is it possible to have a preorder with different kinds of morphisms? Every pair between every object still has only one morphism.
However the morphism in question is always a different one (except the identity morphism).
An informal example: Following David Spivaks [1] Olog approach imagine a hungry dog that always eats the food bought by its owner. 
Assume three objects: A Dogowner O, a Dog D and Dog Food F.
Additionally assume four morphisms: "owns" , "eats", "buys" (short for "buys dog food" which is the only thing he buys) and "is". 
Each of the object are themselves, so there is an "is" morphism from each object to itself.
Accordingly, D "owns" O, O "eats" F, and D "buys" F.
Finally the dog is by definition always hungry and eats all the food given to it
so it should hold that 
"owns" o "eats" =  "buys".
The question in this case: Would this be a preorder?
It fulfills all the criteria for a category: Identity morphisms and compositionality are given.
Following [2] it does also fulfill the criteria for a preorder that
"a proset is a (strict) thin category: a strict category such that for any pair of objects x, y, there is at most one morphism from x to y."
However I have not seen anything similar in any of the usual examples: ⊆ and  ≤ are the usual examples for preorders and are the only morphisms applied to the objects in the category.
Best Regards
Pavel
PS: I couldn't come up with a more "formal" example which may be an indicator that I am on the wrong track.
SOURCES:
[1] Spivak, David I,  Robert E. Kent, "Ologs: A Categorical Framework for Knowledge Representation"
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0024274 
[2] https://ncatlab.org/nlab/show/preorder
 A: Yes, this is a preorder.  One of the key insights of category theory is that often, abstract properties of objects (and morphisms between them) are more important than their concrete descriptions.  As you pointed out, the category you have described doesn't "look like a preorder", because the morphisms aren't given names like $\subseteq$ or $\leq$.  But category theory doesn't care about names.  Your category satisfies the definition of a preorder, so e.g. if you had some fancy theorem about preorders, it would be perfectly valid to apply it to this category.
A: The short answer is yes. The names/meanings of the morphisms are not part of the data: what's important is how the morphisms compose. Remember that to define a category is to specify the set of objects, and set of morphisms (along with the identities and composition). (It may be that $\{ \text{owns}, \text{eats}, \text{buys} \}$ isn't, strictly speaking, a well-formed set, because a priori the elements have not been defined, though in general it is harmless to consider it a set, because taking any set of cardinality 3 will be suitable.) You should therefore feel free to label the morphisms as you like: if you forget the names, and just consider the underlying graph along with the structure of composition, you'll see that you have exactly a preorder.
