  We consider the problem of when one quandle homomorphism will factor through
another, restricting our attention to the case where all quandles involved are
connected. We provide a complete solution to the problem for surjective quandle
homomorphisms using the structure theorem for connected quandles of Ehrman et
al. (2008) and the factorization system for surjective quandle homomorphsims of
Bunch et al. (2010) as our primary tools. The paper contains the substantive
results obtained by an REU research group consisting of the first four authors
under the mentorship of the fifth, and was supported by National Science
Foundation, grant DMS-1659123.
