What is the short form of 'Let us'?
Lets or Let's?
Whenever a contraction is made using two words, we always indicate this by using the ' sign to show that part of a word has been omitted:
It is : it's
He is : he's
who is : who's
he can not : he can't
he would not : he wouldn't
Don't confuse let's as above, with lets as in, He has a flat/apartment/house which he lets out by the week or month (ie people can rent this accommodation from him.)