Transporter system induced by a finite group on a finite p-subgroup

Definition
Suppose $$G$$ is a finite group, $$p$$ is a prime number, and $$P$$ is a finite $$p$$-subgroup of $$G$$. The transporter system on $$P$$ induced by $$G$$, which we will denote $$\mathcal{T}_P(G)$$, is a category defined as follows:


 * The objects of the category are the subgroups of $$P$$.
 * The morphisms are defined as follows: for subgroups $$R,S \le P$$, the morphism set is parametrized by $$N_G(R,S) = \{ g \in G \mid gRg^{-1} \le S \}$$. Note that each such $$g \in G$$ gives rise to a set map $$c_{g,R,S}: x \mapsto gxg^{-1}$$ that is a group homomorphism from $$R$$ to $$S$$. However, the set map or the group homomorphism itself does not convey the full information about the categorical morphism, because it is possible for different group elements to induce the same morphism.
 * Composition of morphisms is defined as follows: Suppose $$g \in N_G(R,S)$$ and $$h \in N_G(S,T)$$. Then, the composite of these morphisms is the element $$hg \in N_G(R,T)$$.
 * The identity morphism of $$R$$ is defined as the identity element in $$N_G(R,R)$$.

Relation between the transporter system and the fusion system
Suppose $$G$$ is a finite group, $$p$$ is a prime number, and $$P$$ is a finite $$p$$-subgroup of $$G$$. Consider the following two categories:


 * The transporter system $$\mathcal{T}_P(G)$$
 * The fusion system $$\mathcal{F}_P(G)$$

The object sets of the two categories are identical, but the morphism sets differ. There is a natural "forgetful" functor from $$\mathcal{T}_P(G)$$ to $$\mathcal{F}_P(G)$$ defined as follows:


 * Each object of $$\mathcal{T}_P(G)$$, namely, a subgroup of $$P$$, is sent to the same subgroup of $$P$$, now viewed as an object of $$\mathcal{F}_P(G)$$.
 * The morphism set map is as follows: the element $$g \in N_G(R,S)$$, which is a morphism from $$R$$ to $$S$$ in $$\mathcal{T}_P(G)$$, gets sent to the homomorphism $$\varphi:R \to S$$ given by $$x \mapsto gxg^{-1}$$.