There are many categories whose objects can be consider as categories, intuitively I think there is an embedding. However, the usual 'embedding functor' can only enlarge the category (I think). For example, if $\mathcal{G}$ is a groupoid with only one object, how to 'embed' $\mathcal{G}$ into Grp, both as categories (using functors or other things). Is it possible?
This problem arised when I try to consider Yoneda lemma as generalization of Cayley's theorem. Yoneda lemma only claims that there is a bijection between sets, I'm wondering how to make it become isomorphism between groups using language in category theorem. In this case, we need to get the isomorphism, which is an invertible arrow in Grp, from a fully faithful functor $Y:\mathcal{G} \rightarrow Set^{\mathcal{G}^{op}}$, whose arrow function gives the bijection $hom_\mathcal{G}(\cdot,\cdot)\rightarrow Nat(\mathcal{G}(-,\cdot),\mathcal{G}(-,\cdot))$. Although we can easily see it is an isomporphism by property of functor, how to convert it into Grp? Especially, if we already know $Y:\mathcal{G}_1 \rightarrow \mathcal{G}_2$ is a functor then how to make Y an arrow in Grp?