It passes into the ventral aspect of each branchial arch through an afferent branchial artery. This artery breaks up into the equivalent of a capillary network within the gills where it is oxygenated, and collects into a more dorsal efferent branchial artery. The names afferent and efferent, meaning leading to and leading away, are with reference to the gills. This serves to remind us of the function of the gill arches and the aortic arches they contain. From the efferent branchial arteries the newly oxygenated blood is distributed to the body tissues.
All blood returning to the heart collects into a sinus venosus.
The lungfish Were Goodrich alive today, he would have looked at coelacanths, perhaps the closest available bony fish relatives of amphibians. Lungfish use their lungs for respiration, as well as their gills. The circuit changes can easily be understood as adaptive for these functional needs. Presumably, having lungs lets the fish get by with less arches involved in gill respiration, for even though the arches 2-4 persist as channels taking blood dorsally, they have no respiratory function at all. Both of these arches supply gills, and in addition a connection from the sixth arch is made to the lungs, as a pair of pulmonary arteries.