The fundamental quandle of ribbon concordances

We describe the fundamental quandle of a properly embedded surface $F$ (possibly with boundary) in $\mathbb{R} ^{3}\times I$, and derive its presentation in terms of a motion picture diagram or a CH-diagram of $F$. Our study is based on the topological definition of the fundamental quandle. We prove that a ribbon concordance $C$ from a classical knot $K_1$ to $K_0$ gives rise to an injective quandle homomorphism $Q(K_0)\to Q(C)$ and a surjective quandle homomorphism $Q(K_1)\to Q(C)$.