What kind of pronoun case is their?

The pronoun 'their' is a possessive adjective, a Word that is placed before a noun to describe that noun as belonging to two or more people or things. Example:

The Washingtons have painted their house.

Mark and Mary brought their baby for a visit.
The possessive adjective 'their' is a third person, plural pronoun.

The possessive adjective 'their' can be used to describe a subject noun or an object noun.

The possessive adjective 'their' is placed before a noun to describe that as belonging to two or more people or things spoken about.

Examples:

The Martins live on Maple Street. Their house is on the corner.

• describes the subject noun 'house' belonging to the Martins/

The Martins live on Maple Street. I will show you theirhouse.

• describes the noun 'house', the direct object of the verb 'show'.

The trees are blocking the view but theirshade is nice.

The trees are blocking the view but I enjoy their shade.