Let \$$\underline{2} = \\\{1, 2\\\} \$$ and \$$\underline{3} = \\\{1, 2, 3 \\\} \$$.
These are objects in the category **Set**.
Write down all the elements of the set \$$\textbf{Set}( \underline{2}, \underline{3} ) \$$; there should be nine.

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The category of sets, denoted **Set**, is defined as follows.

(i) \$$Ob(\textbf{Set}) \$$ is the collection of all sets.

(ii) If \$$S\$$ and \$$T\$$ are sets, then \$$\textbf{Set}(S, T) = \\\{ f : S \rightarrow T | f \text{ is a function } \\\} \$$.

(iii) For each set \$$S\$$, the identity morphism is the function \$$id_S : S \rightarrow S \$$ given by \$$id_S (s) = s \$$ for each \$$s \in S \$$.

(iv) Given \$$f : S \rightarrow T \$$ and \$$g : T \rightarrow U \$$, their composite \$$f . g\$$ sends \$$s \in S \$$ to \$$g( f (s)) \in U \$$.