Kerodon

Example 4.7.0.1. When speaking informally, it is common to say that the category $\operatorname{Set}$ has all limits and colimits. A more precise statement is that the category $\operatorname{Set}$ has all small limits and colimits; that is, every diagram $F: \operatorname{\mathcal{J}}\rightarrow \operatorname{Set}$ indexed by a small category $\operatorname{\mathcal{J}}$ has a limit and colimit. Here the size restriction on $\operatorname{\mathcal{J}}$ cannot be omitted. For example, if $\{ S_ j \} _{j \in J}$ is a collection of sets indexed by another set $J$, then it is permissible to form the coproduct ${\coprod }_{j \in J} S_ j$. However, it is not permissible to form the coproduct ${\coprod }_{S \in \operatorname{Ob}(\operatorname{Set})} S$ of all sets.