Yoneda embedding ($y:\mathcal{C}\to [{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$)

The Yoneda embedding sends $A$ to $\mathcal{C}(-,A)$. The Yoneda lemma identifies natural transformations $yA\to yB$ with morphisms $A\to B$, so the embedding is full and faithful. It preserves every limit that exists in $\mathcal{C}$.