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This is a reverse of the question “Is there a finitely complete category with terminal object but NO subobject classifier?” From “An informal introduction to topos theory” by Tom Leinster I learned that there is 3 definitions of a subobject classifier in some category C:

  1. where we directly work with morphisms of C: for every monomorphism there exists a characteristic morphism etc.;
  2. a terminal object in the category of monomorphisms and pullback squares;
  3. the functor Sub is representable.

In order to make sense of (3) we need Sub which is defined via pullbacks in C. (3) requires C to have pullbacks. But (1) and (2) do not, though they imply existence of the terminal object. Is there a category with a subobject classifier and which is not finitely complete? (AFAIK subobject classifier → terminal object → (have pullbacks ↔ have finite limits = is finitely complete).)

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Yes: take the full subcategory of $Set$ whose objects are sets of cardinality 2 or less.

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