I'm reading "an introduction to category theory" by Harold Simmons. In this books, exercise 1.2.7 wants us to show that $\mathcal{Set}_\bot$ (the category of pointed sets) and $\mathcal{Pfn}$ (the category of sets and partial functions) are "essentially the same" category. So I guess this is to let us prove that these two categories can be somehow converted to each other. Here is my attempt:
For every arrow $\phi : S \rightarrow T$ in $\mathcal{Set}_\bot$, just restrict the domain of $\phi$ to $\{\bot_s\}$, and then the category becomes $\mathcal{Pfn}$.
For every arrow $f : S \rightarrow T$ in $\mathcal{Pfn}$, let $X$ be the domain of $f$. For each $x \in X$, we make two pointed sets: $(S,x)$, $(T,f(x))$ and also an arrow between these two pointed sets, which is a function: $\forall x' \in S, f(x') = f(x)$. I think this should convert $\mathcal{Pfn}$ into a $\mathcal{Set}_\bot$.
Obviously there are many ways to convert from $\mathcal{Pfn}$ to $\mathcal{Set}_\bot$ and the other way around might also be true. However, the problem is: once I convert one to the other, some information are "dropped" so that I can't convert it back to recover the category before conversion.
So my question is:
- When we say two categories are "essentially the same", what does it exactly mean?
- Is my attempt valid?
- Is there some way to convert between each other without losing anything?