# Universal property (Riehl):Unique isomorphism which commutes with the chosen representations??

Emily Riehl writes at the beginning of Chapter 2.4 (in Category theory in context) regarding the universal property:

More precisely, there is a unique isomorphism between c and any other object representing F that commutes with the chosen representations.

So just to be sure, what commuting square is she exactly referring to?

My thoughts: We have two universal objects $$c,d$$ and two chosen representations (natural isomorphisms) $$\alpha : C(c, -) \Rightarrow F, \beta : C(d, -) \Rightarrow F$$. And let $$f:c \rightarrow d$$ be an isomorphism. What does she mean now with the condition that it should commute with the chosen representations.

Now i am not to sure what should commute.

First guess: $$\forall k \in C: \beta_k(g) = \alpha_k(gf) = \alpha_k(f^*(g))$$

Second guess: $$Ff(\alpha_c(1_c)) = \beta_d(f) = (\beta \cdot f^*)_d(1_d)$$

Is one of these correct or none and why?

Thanks a lot!

The first idea is correct. Given such $$\alpha$$ and $$\beta$$, you can find a unique isomorphism $$f: d \rightarrow c$$ such that
$$\beta \circ f^{\ast} = \alpha.$$
I do not quite understand your second statement, as by the Yoneda lemma we get $$Ff(\alpha_c(1_c)) = \alpha_d(f)$$, and not equal to $$\beta_d(f)$$ as you state.