One of the most famous TFNP subclasses is PPP, which is the set of all search problems whose totality is guaranteed by the pigeonhole principle. The author's recent preprint [ECCC TR24-002 2024] has introduced a TFNP problem related to the pigeonhole principle over a quotient set, called Quotient Pigeon, and shown that the problem Quotient Pigeon is not only PPP-hard but also PLS-hard. In this paper, we formulate other computational problems related to the pigeonhole principle over a quotient set via an explicit representation of the equivalence classes. Our new formulation introduces a non-trivial PPP$\cap$PPA$_k$-complete problem for some $k \ge 2$. Furthermore, we consider the computational complexity of a computational problem related to König's lemma, which is a weaker variant of the problem formulated by Pasarkar, Papadimitriou, and Yannakakis [ITCS 2023]. We show that our weaker variant is PPAD-hard and is in PPP$\cap$PPA.