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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!

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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.

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