Learn Icelandic: Hún fer í bíóið með vinnufélaganum og mér. - She is going to the cinema with the colleague and me.

The preposition með triggers the dative case when it means "with" in the sense of accompaniment (going together). Because she is going with both of them, both the colleague and "me" must be in the dative case.

How does the word vinnufélagi become vinnufélaganum?

First, the preposition með changes the base word vinnufélagi (colleague) into its dative form, vinnufélaga. Then, because we are talking about "the" specific colleague, we add the dative definite article ending -num, resulting in vinnufélaganum.

Why use mér for "me" instead of mig?

mér is the dative form of ég (I). Since the preposition með requires the dative case here, you have to use mér. mig is the accusative form, so it would be incorrect in this sentence.

Does the word bíóið also change its case in this sentence?