For an integral domain \(A\) we consider the contravariant functor \(\hom(\_, A)\) from the category of sets to the category of commutative unital \(A\)-algebras. We note that since \(A\) is an integral domain the idempotents of \(\hom(L, A)\) for any set \(L\) are precisely the maps from \(L\) to \(A\) with values in \(\{0, 1\}\).

  1. Lemma. \(\hom(\_, A)\) is pseudomonic.

The following example shows that we cannot assume the unrestricted functor \(\hom(\_, A)\) to be full, if \(A\) is a general ring.

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