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lemma. isomorphism to class of subobjects [kostecki2011introduction, 6.7] [tt-004K]
✍️source

    In any category \({\cal C}\) with a subobject classifier \(\Omega \), \[ \operatorname {Sub}(X) \cong {\cal C}(X, \Omega ) \] i.e. the class of subobjects of an object \(X\) in \({\cal C}\) is isomorphic to the class of arrows from \(X\) to \(\Omega \).

    proof. 

      It follows from the definitions and lemma [tt-003J] that for every \(f : Y \to X\) and \([f] \in \operatorname {Sub}(X)\),
      • (surjection) \(\chi (f) \in {\cal C}(X, \Omega )\)
      • (injection) for every \(h : X \to \Omega \), \(\chi (f) = h\) since is a pullback.