Sheaves in Geometry and Logic MacLane and Moerdijk
Chapter 1

Exercise 3
The category of modules over a ring has no subobject classifier

Exercise 4
An equivalence of categories preserves a subobject classifier

Exercise 5
Defining the exponential in monoids and groups (considered as one object categories)

Exercise 8
Rigorous derivation of the exopnential in a presheaf category (completing the proof the presheaf categories are Cartesian closed)

Exercise 9
Characterizing the subobject classifier in the category of presheaves on a poset

coYoneda Lemma
We prove the coYoneda lemma directly  the fact that every presheaf is a canonical colimit of representable presheaves.

Subobject classifer in presheaf category
We construct the subobject classifer in the category of Setvalued presheaves