Definition Weak limits [efr-002B]

Let X: I \to \mathcal {C} be a diagram. A cone A \xrightarrow {f_i} X_i is said to be a weak limit if, for any other cone B, there exists a map B \to A (but this map is not necessarily unique).

Observe that, if an actual limit L exists, a cone is a weak limit if and only if the induced map A \to L admits a section. It follows that any functor which preserves the limit of some diagram also preserves the weak limits (in the sense that they are carried to another weak limit).