This series of posts will assume some knowledge of Category Theory. For a great overview of Category Theory watch the "TheCatsters" lectures on Youtube. Here is a link aggregate to the videos.
The content of the series will loosely follow a section of Toposes and Local Set Theories by John Lane Bell.
Let $\mathbf{C}$ be a category. For $A, B \in Ob(\mathbf{C})$, let $\mathscr{C}(A,B)$ be the set of arrows between $A$ and $B$. Define the functor (also called a bifunctor since it's leaving a product of two categories) $hom_\mathbf{C} : \mathbf{C}^{op} \times \mathbf{C} \rightarrow \mathbf{Set}$ by $hom_{\mathbf{C}}(A,B) = \mathscr{C}(A,B)$ on objects and $hom_{\mathbf{C}}(f^{op}:A_1\rightarrow A_2, g:B_1\rightarrow B_2):\mathscr{C}(A_1,B_1) \rightarrow \mathscr{C}(A_2, B_2)$ by $hom_{\mathbf{C}}(f^{op}, g) = g\_f$, where $f:A_2\rightarrow A_1, g:B_1 \rightarrow B_2$ are arrows in $\mathbf{C}$ and the $\_$ is where the argument would go. Then for fixed $A \in Ob(\mathbf{C})$ we may define functors $H^A: \mathbf{C}^{op} \rightarrow \mathbf{Set}, H_A:\mathbf{C} \rightarrow \mathbf{Set}$ by $H^A = hom_{\mathbf{C}}(\_,A), H_A = hom_{\mathbf{C}}(A,\_)$ on objects and $H^A = hom_{\mathbf{C}}(\_,1_A), H_A = hom_{\mathbf{C}}(1_A,\_)$ on arrows. Then $H^A, H_A$ are members of the functor categories $\mathbf{Set}^{\mathbf{C}^{op}}, \mathbf{Set}^{\mathbf{C}}$, respectively. The arrows in functor categories are natural transformations.
It's useful to think of $hom_\mathbf{C}$ as an "inner product on $\mathbf{C}$" in the sense that "conjugate linear in first argument" $\leftrightarrow$ covariant in first argument and "linear in second argument" $\leftrightarrow$ covariant in second argument. That is $hom_\mathbf{C} : \mathbf{C}^{op} \times \mathbf{C} \rightarrow \mathbf{Set} \leftrightarrow <\_,\_>:\mathbf{H}^* \times \mathbf{H} \rightarrow \mathbb{C}$. Moreover to continue the Hilbert space analogy, $H^A \leftrightarrow <\_,x>:\mathbf{H}^*\cong \mathbf{H}^{op} \rightarrow \mathbf{C}$ and $H_A:\leftrightarrow <x,\_>:\mathbf{H} \rightarrow \mathbb{C}$. So, in particular, under this dictionary, $H^A \leftrightarrow <\_,x>$. In Hilbert space theory, we have by the Riesz theorem that $\mathbf{H} \cong \mathbb{C}^{{\mathbf{H}}^*}$ under the map $x\mapsto <\_,x>$ that, in particular, embeds $\mathbf{H}\hookrightarrow \mathbb{C}^{{\mathbf{H}}^*} \cong \mathbb{C}^{{\mathbf{H}}^{op}}$. Analogously, the Yoneda Embedding Theorem is the result that $\mathbf{C} \hookrightarrow \mathbf{Set}^{\mathbf{C}^{op}}$. This statement will be made precise in the next post and the prerequisite Yoneda Lemma will be proven.
