Weak enriched Yoneda lemma

Question

I am reading http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf and on pg. 21, for the proof of the weak enriched Yoneda lemma starting from "Next, if α is defined by (1.47)..." he wants to show that a certain composite is the identity. I can't seem to verify this rigorously! Why is it the identity? He mentions something about $j_I=i_I:I \rightarrow [I,I]$, but I don't see how that comes into play. Any help would be appreciated!

Answer 1

So, I solved this one but I don't see what the comment about $I\rightarrow [I,I]$ has to do with anything. For further reference, and so that people, if they feel up to it can check my reasoning I will answer my own question.

Let us change notation compared somewhat to what Kelly uses in his paper. Let $\ast$ be the unit object, and let $\underline{D}$ be our category, enriched over $\underline{\mathcal{V}}$. Then, say that we have a morphism $f:\ast \rightarrow Fd$ for a functor $F: \underline{D} \rightarrow \underline{\mathcal{V}}$. This morphism corresponds under adjunction and precomposition with the isomorphism $\ast \times \ast \rightarrow \ast$ to a map $f^\ast:\ast \rightarrow \underline{\mathcal{V}}(\ast,Fd)$.Now, one checks that the morphism $\mathcal{V}(f,1)$ that Kelly refers to is just precomposition with this morphism. Now, through some tedious diagram chasing one readily verifies that the claimed composite is $f$, using among other things the counit-units to see that the composite reduces to $ev(1 \times f^\ast)l_*$ where $l_*: \ast \rightarrow \ast \times \ast$ is the unit iso. By the counit-unit conditions, $ev(1 \times f^\ast) = fl_*^{-1}$ and so the composite reduces to $f$.