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Hi, in Category of elements, it says:
It is the category whose objects are pairs $(c,x)$ where $c$ is an object in $C$ and $x$ is an element in $P(c)$ and morphisms $(c,x) \to (c',x')$ are morphisms $u:c \to c'$ such that $P(u)(x) = x'$.
S. Awodey in his notes here in his definition (actually in proposition 8.10) defines the concept in the same terms, but he adds:
actually, the arrows are triples of the form $(u,(x,C),(x',C'))$ satisfying…
And would be nice to have that here also (that the arrows are those triples). If not technically the same arrow could belong to two hom-sets.
My own feeling is that this is merely a slight and forgivable abuse of language; for me it is implicit that the domain and codomain are always part of the data of a morphism, and the $u$ names the only extra datum needed to complete the description.
In other words, we generally use this definition of a category in which it doesn’t matter whether or not the hom-sets are disjoint. If you want a structured way to make the hom-sets disjoint, see protocategory.
However, this is a good example of that point, so perhaps it should be brought out more explicitly.
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