In page 493 of Paolo Aluffi's Algebra: Chapter 0, it says (Lemma 1.17):
If $\mathcal G: D \to C$ has a left-adjoint $\mathcal F: C\to D$ and $\mathcal A: I\to D$ is another functor, then there is a canonical isomorphism $$\mathcal G(\lim \limits_{\gets} \mathcal A) \simeq \lim \limits_{\gets}(\mathcal G \circ \mathcal A)$$ (if the limits exist).
I have two questions on this lemma:
In Aluffi's book, it seems to be one of the conditions that both $\lim \limits_{\gets} \mathcal A$ and $\lim \limits_{\gets}(\mathcal G \circ \mathcal A)$ exist. I wonder whether the existence of one of the limits implies the existence of the other limit.
I believe by 'canonical', it means the isomorphism is somewhat a natural transformation. However, I can't figure out how exactly it can be seen as a natural transformation.