Summary: 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 G. Ehrman et al. [J. Knot Theory Ramifications 17, No. 4, 511–520 (2008; Zbl 1148.57018)] and the factorization system for surjective quandle homomorphisms of E. Bunch et al. [J. Knot Theory Ramifications 19, No. 9, 1145–1156 (2010; Zbl 1210.57013)] as our primary tools.