My argument is as follows :

(i) \\(f◦g\\) is a morphism from \\(d \to d \\). Since the only morphism in \\(C \\) from \\(d \to d \\) is \\(id_d \\) we must have \\(f◦g= id_d \\).

(ii) \\(g◦f\\) is a morphism from \\(c \to c \\). Since the only morphism in \\(C \\) from \\(c \to c \\) is \\(id_c \\) we must have \\(g◦f= id_c \\).

(i) \\(f◦g\\) is a morphism from \\(d \to d \\). Since the only morphism in \\(C \\) from \\(d \to d \\) is \\(id_d \\) we must have \\(f◦g= id_d \\).

(ii) \\(g◦f\\) is a morphism from \\(c \to c \\). Since the only morphism in \\(C \\) from \\(c \to c \\) is \\(id_c \\) we must have \\(g◦f= id_c \\).