We’ve worked with Coursera to create new software for you to assess each other’s work.
Isn't this supposed to be "for you to assess each other's work with?" Like, I have a pen to write with.
Because this second "to" explains the software.
What do you think?
The fact that it's possible to append the preposition with in OP's example doesn't mean it's necessary.
In the syntactically identical "(You) have eyes to see (with)", clearly we don't normally include with...
Sooner or later you'll probably come across the completely spurious "rule" that you should never end a sentence with a preposition. The votes on that link (to an ELU answer on the topic) show that everyone agrees the rule itself is rubbish, but it's worth making the point that there's a grain of truth in there...
1: I have no money to pay you with.
2: I have no money to pay the constant stream of unlicensed carol singers every Christmas with.
In #1, although it's not actually necessary, with is perfectly acceptable (there are a couple of hundred written instances in the link). But most native speakers wouldn't find #2 anywhere near so acceptable.
Both constructions involve the underlying activity to pay with money. We can introduce an indirect object (to pay someone with money), but if that indirect object is a very long noun phrase, it just becomes too cumbersome to "remember" what the preposition refers back to. So we either discard it completely, or resort to a rephrasing such as I have no money with which to pay [some very long noun phrase].
The sentence is correct. The complement of the preposition for is a verbal phrase.
The second to isn't really a preposition of its own. It's part of the verbal phrase for [subject] to [verb], which means “with the purpose that [subject] [verb]”. An equivalent, grammatical way to phrase this clause would be “… software whose purpose is that you can assess each other's work” (this is idiomatic in that this sentence doesn't sound wrong, but it's too long-winded for normal use).
The clause that explains the software is the one introduced by for. If there was no subject in that clause, then we could use either for + gerund (“… software for assessing students' work”) or to + infinitive (“… software to assess students' work”). Since the subject is different, we have to use for followed by the subject (in the object case: “… for them to …”), followed by to and the verb.
“… for you to assess each other's work with” is grammatically correct, but not really idiomatic. It would be a way to combine “We've worked with Coursera to create new software” and “The purpose of what we're doing is that you can assess each other's work with this software”. Here with isn't a particle that's attached to the verb like in go to, and there is no complement of the preposition hidden in the clause like in “I have a pen to work with” (“new software” is part of the previous clause), so with is superfluous and would normally not be used. The fact that the assessment is made with the new software is semantically implied.
The sentence could be presented as:
We (a group of people working with Coursera) created new software for you [that allows you] to assess each other's work.
How it is written in the question implies the "with" that you have mentioned.
