I've been struggling to write a complete proof following the guide provided in [1]. As showed in the book, I constructed the potential isomorphisms $\alpha:Y^{T_1 +T_2}\rightarrow Y^{T_1}\times Y^{T_2}$ and $\beta:Y^{T_1}\times Y^{T_2}\rightarrow Y^{T_1+ T_2}$. For details and notation, please refer to the following link: Construction of potential isomorphisms (For some reason, equations are not showing on the direct link, please try opening in google docs which seem to work fine).
My strategy was to show that $$e_Y\circ (1_{T_1+ T_2}\times (\beta \circ \alpha))=e_Y$$ (where $e_Y$ is the evaluation map $(T_1 + T_2)\times Y^{T_1+ T_2}\rightarrow Y$
which I attempted as following:
By using distributive property of $\times$ over $\circ$, $$e_Y\circ (1_{T_1+ T_2}\times \beta)\circ (1_{T_1+ T_2}\times \alpha)=e_Y$$
Now substituting $e_Y\circ (1_{T_1+ T_2}\times \beta)=\beta_p$ from step 9 of $\beta$'s construction.
$$\beta_p \circ (1_{T_1+T_2}\times \alpha)=e_Y$$
Substituting the definition of $\beta_p$
$$e''_Y \circ (1_{Y^{T_1} \times Y^{T_2}}\times (\delta_1 + \delta_2))\circ \gamma \circ ((1_{T_1+ T_2}\times \alpha)=e_Y$$
where $\delta_1 = \lceil e_{Y_1} \circ (1_{T_1} \times p_1) \circ \gamma'_1 \rceil$ and $\delta_2 = \lceil e_{Y_2} \circ (1_{T_2} \times p_2) \circ \gamma'_2 \rceil$.
This is where I got stuck. I tried using the distributive property (which is not yet introduced at the point in the book. It is right after this exercise) to distribute $1_{Y^{T_1} \times Y^{T_2}}$ over $(\delta_1 +\delta_2))$ because then I could use step 5 from $\beta$'s construction.
Thanks in Advance. Any kind of help is much appreciated.
References:
[1] F. William Lawvere and Stephen H. Schanuel's Conceptual Mathematics.
Though it's possible to manipulate the morphisms directly, it's much easier to work with the universal properties at the level of natural isomorphisms of hom-sets, as in the following. I've left out the justification for each step: see if you can see why each step is justified, and why this implies the property you are trying to establish. (I'm going to use the standard notation for coproduct and product, for anyone coming across this question in the future.)
\begin{align}\mathscr C(X, Y^{T_1 + T_2}) & \cong \mathscr C(X \times (T_1 + T_2), Y) \\ & \cong \mathscr C(X \times T_1 + X \times T_2, Y) \\ & \cong \mathscr C(X \times T_1, Y) \times \mathscr C(X \times T_2, Y) \\ & \cong \mathscr C(X, Y^{T_1}) \times \mathscr C(X, Y^{T_2}) \\ & \cong \mathscr C(X, Y^{T_1} \times T^{T_2})\end{align}
(Note: I'm not quite sure where this exercise appears in the book, so I'm not sure whether any of the above will be new. Let me know if anything is unclear, and I will update my answer.)