Show the following arrows are transposes of each other

Question

Within a cartesian closed category;

Given $Z\xrightarrow{g}W\xrightarrow{f} Y^X$, we want to show that the transpose of $f\tiny{\circ}$$g$ is the following:

$Z\times X\xrightarrow{g\ \times 1_X}W\times X\xrightarrow{\bar{f}} Y$, where $\bar{f}$ is the transpose of $f$.

I tried making use of the universal property of exponentials, but end up in quite a mess. Thus I feel like I'm doing something wrong. How do I show this as cleanly as possible ?

Cheers

Answer 1

Consider the diagram below:

By definition (writing morphism composition in diagrammatic order):

• $\bar f=(f\times 1_X)\varepsilon^X_Y:W\times X\to Y$ is the transpose of $f$
• $\bar h=((gf)\times 1_X)\varepsilon^X_Y:Z\times X\to Y$ is the transpose of $gf$

Consequently: \begin{align} \bar h &=((gf)\times 1_X)\varepsilon^X_Y\\ &=(g\times 1_X)(f\times 1_X)\varepsilon^X_Y\\ &=(g\times 1_X)\bar f\\ \end{align}