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 \$$.