Tensor products of profinite groups

Question

Let $R$ be a profinite commutative ring, and let us take two ordinary $R$-modules, $M$ and $N.$Let $M^\wedge, N^\wedge$ be the profinite completion as $R$-modules, i.e we complete with respect to the submodules such that the quotient is finite. Consider the completed tensor product $M^\wedge \hat{\otimes}_R N^\wedge.$ Is it true that $M^\wedge \hat{\otimes}_R N^\wedge \cong (M \otimes_R N)^\wedge,$ i.e that the completed tensor product commutes with profinite completion?