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1. • CommentRowNumber1.
   • CommentAuthorjesuslop
   • CommentTimeJul 25th 2018
   • (edited Jul 25th 2018)
   • PermaLink
   Author: jesuslop
   Format: MarkdownItexHi, in [Category of elements](https://ncatlab.org/nlab/show/category+of+elements#definition), 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](https://www.andrew.cmu.edu/course/80-413-713/) 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 25th 2018
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   Author: Todd_Trimble
   Format: MarkdownItexMy 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJul 25th 2018
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   Author: Mike Shulman
   Format: MarkdownItexIn other words, we generally use [this definition of a category](https://ncatlab.org/nlab/show/category#AFamilyOfCollectionsOfMorphisms) 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]].

   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.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJul 25th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexHowever, this is a good example of that point, so perhaps it should be brought out more explicitly.

   However, this is a good example of that point, so perhaps it should be brought out more explicitly.