Group Theory/Group objects and torsors

Definition (group object):

Let ${\mathcal {C}}$ be a category that has a terminal object, which we shall denote by $I$. A group object of ${\mathcal {C}}$ consists of

1. an object $G$ of ${\mathcal {C}}$ such that the categorical products $G\times G$, $G\times (G\times G)$ and $(G\times G)\times G$ exist
2. a morphism $\mu :G\times G\to G$, thought of as the multiplication morphism,
3. a morphism $\eta :T\to G$, thought of as the constant identity map (even though $G$ may not even be a set)
4. and a morphism $\iota :G\to G$, thought of as the inversion morphism

which satisfy the following equations:

1. $\mu \circ (\iota ,\operatorname {Id} _{G})=\mu \circ (\operatorname {Id} _{G},\iota )=\operatorname {Id} _{G}$ (identity law)
2. $\mu \circ (\operatorname {Id} _{G},\iota )\circ \Delta =\mu \circ (\iota ,\operatorname {Id} _{G})\circ \Delta =\eta \circ t$, where $\Delta$ shall denote the diagonal of $G$ and $t:G\to T$ is the unique morphism given by the definition of a terminal object (inverse law)
3. $\eta \circ (\eta ,\operatorname {Id} _{G})=\eta \circ (\operatorname {Id} _{G},\eta )\circ \alpha$, where $\alpha :(G\times G)\times G\to G\times (G\times G)$ is the canonical isomorphism (associative law)

Here, we wrote $(f,g):A\times B\to C\times D$ for the morphism that is induced by two maps $f:A\to C$ and $g:B\to D$ and the universal property of the product, by specifying that the map from $A\times B$ to $C$ shall be given by $f\circ \pi _{A}$, and that the map from $A\times B$ to $D$ shall be given by $g\circ \pi _{B}$.

Definition (action of a group object):

Let ${\mathcal {C}}$ be a category which possesses a terminal object, and let $G$ be a group object of ${\mathcal {C}}$. Let $X$ be another object of ${\mathcal {C}}$. An action of $G$