is $Hom(P,N \otimes_{End(P)} P) = N$?

Question

This is probably well known to people who work with algebras but I couldn't find a reference.

Say I have a ring A and a module P and I take B = End(P), the endomorphism ring. Let N be a B-module, is it true that

$Hom_A(P, N\otimes_B P) = N$?

maybe I want to assume for N and P to be projective (over B and over A respectively).

Is this still true if now A is a dg-algebra (or simplicial algebra or ring spectrum or...) and in the identity above everything is derived?

The reason I think it's plausible is because morally

$$ Hom_A(P, N \otimes_B P) = (N \otimes_B P) \otimes_A P^\vee = N \otimes_B B $$

but I want to make sure I haven't "cheated" with the identity above.

thanks