I thought that, “ Nous sommes en train de le finir.” is incorrect since de le becomes du…
Just to add to what Jim said, it is to do with what 'le' represents. If it is a definite article and attached to a noun it will contract.
As in those examples -
C'est le chien de le voisin ----> C'est le chien du voisin
Le problème de le garcon -----> Le problème du garçon
Hope this helps!
To write "Nous sommes en train de le finir" is valid because "le" is the direct object of the verb finir.
You are expressing "We are in the process of finishing it" (to finish it)
I think that you are confusing grammatical articles with a direct object pronoun.
Very understandable confusion -- the grammar can be very tricky.
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