The <inverse limit> is the submodule of the direct product consisting of compatible families:
$$
\varprojlim_{i\in I}N_i
=\left\{(x_i)\in\prod_{i\in I}N_i:
g_{ij}(x_j)=x_i\text{ whenever }i\le j\right\}.
$$
Its projection to $N_i$ sends $(x_j)$ to $x_i$; these projections satisfy the <universal property of an inverse limit>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 101 / 4 / a / Solution

Maps from a module $X$ to $\varprojlim_iN_i$ correspond naturally to compatible families of maps $X\to N_i$.


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 Incoming links: Universal property of an inverse limit