Does the sun sun traces shortest path across local sky on june solstice

The June solstice in the Northern hemisphere is the summer solstice. The June Solstice in the Southern hemisphere is the winter solstice. The summer solstice is equivalent to the longest day while the winter solstice is equivalent to the shortest day. Therefore on the local sky, when is the June solstice we have have the longest day (longest path of sun in the sky) in the Northern hemisphere and the shortest day (shortest path of sun in the sky) in the Southern hemisphere.