Let lambda>kappa and coll(kappa, lambda) be the poset collapsing lambda to kappa. Pick a subposet P which is <kappa-closed and of size lambda. Can we say that P is forcing equivalent to coll(kappa,lambda)? Or, more generally, which are the minimal conditions to make P equivalent to coll(kappa,lambda)?

When lambda=kappa (i.e., kappa cohen forcing) I am pretty sure that this be true, so I was interested in understanding whether it generalizes.