\(\mathbf {Set}\), the category of sets, consists of objects which are sets, and arrows which are functions between them. The axioms of composition, associativity and identity hold due to standard properties of sets and functions.
\(\mathbf {Set}\) has the initial object \(\varnothing \), the empty set, and the terminal object, \(\{*\}\), the singleton set.
\(\mathbf {Set}\) doesn't have a null object.
Monic arrows in \(\mathbf {Set}\) are denoted by \(f: X \hookrightarrow Y\), interpreted as an inclusion map (see also inclusion function in nLab).
Given \(X : \mathbf {Set}\), the subobjects of \(X\) are in canonical one-to-one correspondence with the subsets of \(X\).