Actually, just thinking on it, you can give a really nice formula for cardinality, \$$\mathrm{Card} : \mathbf{Set} \to \mathbb{N} \$$,

\$\mathrm{Card}(X) = \sum_{x \in X}1, (X \in Obj(C)), \$
and,
\$\mathbb{N}(\mathrm{Card}(X), \mathrm{Card}(Y)) = \prod_{y \in Y}\sum_{x \in X}1. \$