Let $A$ be a commutative ring and $I$ an index set. Let $(M_i, f_i)_{I\in I}$ be an inverse system of $A$-modules and let $N$ be an $A$-module. There is a canonical map
$$\phi: (\mathrm{lim}_{\leftarrow i}M_i)\otimes_AN\to\mathrm{lim}_{\leftarrow i}(M_i\otimes_AN)$$
given by $\sum_j(x_i)_{I\in I}^{(j)}\otimes y^{(j)}\mapsto(\sum_jx_i^{(j)}\otimes y^{(j)})_{i\in I}$, where the sum over $j$ is finite.
Why is $\phi$ not surjective? The connecting morphisms in $(M_i\otimes_AN)$ are given by $f_i\otimes1$ which means that we cannot have elements of the form $(\sum_jx_i^{(j)}\otimes y_i^{(j)})_{i\in I}$. So why is this map not surjective. It looks surjective.
Why is $\phi$ not injective? Does it become injective when $N$ is a finitely generated free $A$-module?