The Styloglossus, the shortest and smallest of the three styloid muscles, arises from the anterior and lateral surfaces of the styloid process, near its apex, and from the stylomandibular ligament.

Passing downward and forward between the internal and external carotid arteries, it divides upon the side of the tongue near its dorsal surface, blending with the fibers of the Longitudinalis inferior in front of the Hyoglossus

Hyoglossus

The Hyoglossus, thin and quadrilateral, arises from the side of the body and from the whole length of the greater cornu of the hyoid bone, and passes almost vertically upward to enter the side of the tongue, between the Styloglossus and Longitudinalis inferior....

; the other, oblique, overlaps the Hyoglossus and decussates with its fibers.

Innervation

The styloglossus is innervated by the Hypoglossal nerve

Hypoglossal nerve

The hypoglossal nerve is the twelfth cranial nerve , leading to the tongue. The nerve arises from the hypoglossal nucleus and emerges from the medulla oblongata in the preolivary sulcus separating the olive and the pyramid. It then passes through the hypoglossal canal...

 (CN XII) like all muscles of the tongue except palatoglossus which is innervated by the Pharyngeal plexus of vagus nerve

Pharyngeal plexus of vagus nerve

The pharyngeal plexus is a network of nerve fibers innervating most of the palate, larynx, and pharynx.It is located on the surface of the middle pharyngeal constrictor muscle.-Sources:...

(CN X).

Action

The styloglossus draws up the sides of the tongue to create a trough for swallowing. As a pair they also aid in retracting the tongue.