I am interested in the category-theoretic description of trees (and operads?) and have started a course of study that will allow me to engage with these two (1, 2) manuscripts of Joachim Kock.

An essential prerequisite to the early portions of the manuscript involves an understanding of a pair of adjoint functors. The left adjoint is the change of basis functor $g^{ast}: mathbf{Set}_{/A} rightarrow mathbf{Set}_{/B}$ associated with the change of basis between sets $g: B rightarrow A$. I find it very easy to reason about the image of a bundle under the change of basis functor. This functor takes a bundle $f: X rightarrow A$ to the pullback of $f$ by $g$, which I can easily interpret in $mathbf{Set}_{/B}$ as the fibered product with the canonical projection $h: X times_{A} B rightarrow B$ given by $h: (x, b) mapsto b$.

Kock utilizes the right adjoint, which I have read is called the dependent product:

$$

(g^* dashv prod_g)

colon

mathbf{Set}_{/B}

stackrel{overset{g^* }{leftarrow}}{underset{prod_g}{to}}

mathbf{Set}_{/A}

,.

$$

I am trying to understand this functor, but am finding it very difficult. Are there any set-theoretic descriptions of the image of a bundle $f: X rightarrow B$ under $prod_g$ that would help me in this context?

Although I’ve only had a little exposure to it, the process of re-describing adjoint functors in the “internal language of their categories” (I hesitate to use this phrase here– I have seen it used in the literature of toposes and do not know its formal definition– perhaps I’ll just call this process *reifying*?) has proven very difficult for me. Are there any mental algorithms that offer any help? Are there a set of useful exercises I can undertake to develop the skill? Are there theorems I can study that will provide insight into the process?

It has struck me in the course of mulling over the problem that I would also like to know whether category-theorists even bother with reifying their constructions. Does this process have a name? Is it done often?