If $\mathcal{C}$ is a category, write $\hat{\mathcal{C}}$ for the category of contravariant functors from $\mathcal{C}$ to ${\bf Sets}$ the category of sets. The morphisms in $\hat{\mathcal{C}}$ are natural transformations of functors.

(To avoid set theoretical concerns, one can take a universe $\mathcal{U}$ and take all categories to be $\mathcal{U}$ small.)

For any object $X$ of $\mathcal{C}$ $h_X = {\rm Hom}(-,X)$ is a contravariant functor from $\mathcal{C}$ to ${\bf Sets}$ and therefore is an object of $\hat{\mathcal{C}}$

Yoneda Lemma says that $X\mapsto h_X$ is a covariant functor $\mathcal{C}\to\hat{\mathcal{C}}$ which embeds $\mathcal{C}$ faithfully as a full subcategory of $\hat{\mathcal{C}}$ This embedding is called the Yoneda embedding.