# Is there a standard name for a category all of whose contravariant hom functors are sheaves?

What prompted this question is the definition of a pseudogroup in nlab:

Given a X a topological space. Then a pseudogroup is a subgroupoid of the groupoid of transitions between open sets in X, contains the groupoid of identity transitions, and satisfies a sheaf condition.

(Pseudogroups of continuous/smooth transitions are used to define the atlases for manifolds of the respective kind).

It seems to me a pseudogroup is morally a groupoid G that satisfies the sheaf condition for each presheaf G[-,V] for V an object of G.

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You need more data than just a category to define what a sheaf is. See ncatlab.org/nlab/show/site . – Qiaochu Yuan Aug 27 '12 at 6:54
What do you mean by a sheaf on arbitrary category? (One needs a Grothendieck topology on category to speak about sheaves. Well, any category has a Grothendieck topology in which all representable functors are sheaves.) – Grigory M Aug 27 '12 at 6:54
So a generalisation of a pseudogroup would be, G a groupoid; I turn G into a site by equipping it with a coverage J (that generates a grothendieck topology), then require J to be subcanonical so that every representable hom functor G[-,V] is a sheaf? – Mozibur Ullah Aug 27 '12 at 10:26