  We study and compare two factorisation systems for surjective homomorphisms
in the category of quandles. The first one is induced by the adjunction between
quandles and trivial quandles, and a precise description of the two classes of
morphisms of this factorisation system is given. In doing this we observe that
a special class of congruences in the category of quandles always permute in
the sense of the composition of relations, a fact that opens the way to some
new universal algebraic investigations in the category of quandles. The second
factorisation system is the one discovered by E. Bunch, P. Lofgren, A. Rapp and
D. N. Yetter. We conclude with an example showing a difference between these
factorisation systems.
