remark. presheaf and Yoneda lemma [leinster2016basic, 4.2.1] [tt-002V]

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The Yoneda lemma says that for any \(X \in {\cal C}\) and presheaf \(\mathscr {F}\) on \({\cal C}\), a natural transformation \(\mathscr {H}_X \to \mathscr {F}\) is an element of \(\mathscr {F}(X)\) of shape \(\mathscr {H}_X\).

We may ask the question [chen2016infinitely, 68.6.4]:

What kind of presheaves are already "built in" to the category \({\cal C}\)?

The answer by the Yoneda lemma is, the Yoneda embedding \(\mathscr {H}_{\bullet }: {\cal C} \to [{\cal C}^{op}, \mathbf {Set}]\) embeds \({\cal C}\) into its own presheaf category.

In mathematics at large, the word "embedding" is used (sometimes informally) to mean a map \(i: X \to Y\) that makes \(X\) isomorphic to its image in \(Y\), i.e. \(X \cong i(X)\). [leinster2016basic, 1.3.19] tells us that in category theory, a full and faithful functor \(\mathscr {I}: X \to Y\) can reasonably be called an embedding, as it makes \(X\) equivalent to a full subcategory of \(Y\).

So, \({\cal C}\) is equivalent to the full subcategory of the presheaf category \([{\cal C}^{op}, \mathbf {Set}]\) whose objects are the representables.