For an <inverse system> of sets $(X_j,f_{ij})$ indexed by a <directed set> $J$, the <inverse limit> is the set of compatible tuples
$$
\varprojlim_{j\in J}X_j
=\left\{(x_j)\in\prod_{j\in J}X_j:f_{ij}(x_j)=x_i\text{ whenever }i\leq j\right\}.
$$
Its projection to $X_j$ sends $(x_i)_i$ to $x_j$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 1 / a / i / Solution

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A directed set is a nonempty <preorder> in which every finite subset has an upper bound. It provides an index set in which any finite collection of stages has a common later stage.

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