A groupoid object in an $(\infty,1)$ category $\mathcal{C}$ is a functor $G:N(FinSet)^{op} \to \mathcal{C}$ such that for any partition $[n]=S \cup S'$ intersecting in $s$, the object $G([n]$ is the pullback of $G(S)$ and $G(S')$ over $G(s) \simeq G([0])$. In particular, $G$ is a group object if $G([0])$ is contractible.

Can anyone explain to me why this definition agrees intuitively with our notion of group objects in ordinary categories? For example, where do "inverses" come from? Can we consider all group objects as monoids in some monoidal category (if the monoid structure comes from products), i.e. is the functor $G$ lax monoidal?

A groupoid object in an $(\infty,1)$ category $\mathcal{C}$ is a functor $G:N(\Delta)^{op} \to \mathcal{C}$ such that for any partition $[n]=S \cup S'$ intersecting in $s$, the object $G([n]$ is the pullback of $G(S)$ and $G(S')$ over $G(s) \simeq G([0])$. In particular, $G$ is a group object if $G([0])$ is final.

Can anyone explain to me why this definition agrees intuitively with our notion of group objects in ordinary categories? For example, where do "inverses" come from? Can we consider all group objects as monoids in some monoidal category (if the monoid structure comes from products), i.e. is the functor $G$ lax monoidal?