Suppose we have a category $C$ such as $hom(C) \subset ob(C)$ (so every arrow $f: a \rightarrow b$ of $C$ is also an object of $C$). What is the name and the main properties of such category?
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I do not think there is a name for the property that you mention, for the simple reason that this property is not very meaningful.
Look at it this way:
Suppose you have a category $C$ which has the property that you mention, and then another category $D$ which has exactly the same objects, but the morphisms are written with red ink (just to enphasize some super-silly difference in naming) or have an overbar, if you prefer.
Now, we certainly agree that $C$ and $D$ should have the same mathematical properties, since they are effectively the same category, yet $C$ has your property, while $D$ does not, since its morphisms are written in red (or have an overbar) and the elements of $Ob(D)$ are all still in black (and have no overbar). So your property is very fragile and not respected even by isomorphic categories.
A mathematical property - to be considered meaningful - needs to respect equivalence of categories and even more isomorphism of categories. Yours does not. This is called the principle of equivalence in nlab
In his comment above @Oskar mentions the more interesting self-enriched categories, which is perhaps what you were looking for.
By the way, the collection of morphisms of $C$ is called $Mor(C)$
This idea just came to my mind recently!
I haven't found anything about it, but wrote a short note, where I called such a thing a 'folded category', and I wanted to model infinite dimensional categories by them, starting out from an unbiased version of bicategories.
Then I posted it on MathOverflow, but nobody answered.
Finally, I found out that some elementary feature seems missing: namely I had to add one more composition for second dimension, but I kept on hoping that it will already generate all we need. Now, I think, there is no way to escape from introducing one more operation for each dimension, and with that, it seems that things get the same (or even more) complicated as with any original naive approach... :(