This word has got a bit of everything that makes the language tricky to learn for monolingual English speakers.

Like, what even is the word stem here?

It's the verb "mutat" ("to point/to show"), which is the meat in a sandwich of affixes.

(In fact etymologically it is quite possible that the origin of this word is just "mut" with "-at" itself being a suffix.)

(It's important to note here that "bemutat" can also mean "to give someone the finger", which spawned the classic joke:

"Hi Steve, haven't seen you in ages, who's this lovely lady with you?"
"Oh, she's my wife."
"And aren't you gonna introduce me to her?"
"Of course, how rude of me. [turns to wife] Fuck you, Joan!")

But let's turn our attention to the "-koz" suffix, cos it's a right weird one. Its purpose is to turn a non-reflexive verb into a reflexive one, fixing the object to the subject. Thus eliminating the possibility of the hilarious but embarrassing misunderstanding in the joke above.

(It can also be used to create reciprocal verbs, as in "szeret -> szeretkezik" ("to love" -> "to make love", which is supposed to be a reciprocal activity. At least linguistically.)